Design Principles for Grain Boundaries in Solid‐State Lithium‐Ion Conductors

Lithium dendrite formation and insufficient ionic conductivity remain primary concerns for the utilization of solid‐state batteries. Given that the interpretation of experimental results for polycrystalline solid electrolytes can be difficult, computational techniques are invaluable for providing insight at the atomic scale. Here, first‐principles calculations are carried out on representative grain boundaries in four important solid electrolytes, namely, an anti‐perovskite oxide, Li3OCl, and its hydrated counterpart, Li2OHCl, a thiophosphate, Li3PS4, and a halide, Li3InCl6, to develop the first generally applicable design principles for grain boundaries in solid electrolytes for solid‐state batteries. The significantly different impacts that grain boundaries have on electronic structure and transport, ion conductivity and correlated ion dynamics are demonstrated. The results show that even when grain boundaries do not significantly impact ionic conductivity, they can still strongly perturb the electronic structure and contribute to potential lithium dendrite propagation. It is also illustrated, for the first time, how correlated motion, including the so‐called paddle‐wheel mechanism, can vary substantially at grain boundaries. These findings reveal the dramatically different behavior of solid electrolytes at the microscale compared to the bulk and its potential consequences and benefits for the design of solid‐state batteries. These design principles are expected to aid the synthesis and engineering of solid electrolytes at the microscale for preventing dendrite propagation and accelerating ion transport.


Introduction
Solid-state lithium-ion batteries promise high energy density and excellent safety but there are many challenges to overcome before DOI: 10.1002/aenm.202301114 these goals can be realized. Crucial to the operation of solid-state batteries is the solid electrolyte, which must be able to compete with more traditional liquid electrolytes with regards to ionic conductivity. [1][2][3][4][5] Solid electrolytes are often produced through the sintering of powders, yielding structures in which grain boundaries (GBs) are highly prevalent. [6,7] GBs should be expected to have profound impacts on the performance of a device throughout its lifetime, some of which can be positive, such as the observed increase in ionic conductivity of Li 3 PS 4 containing nanosized grains. [8] However, GBs are generally considered to be performance bottlenecks in solid electrolytes and can cause decreased ionic conductivity [9,10] and even device failure through the propagation of lithium dendrites, leading to short circuits. [11,12] Clearly, an in-depth understanding of the impact of GBs on solid electrolytes is essential, but given that there is a huge variety of candidate electrolyte materials, [13][14][15] this is a not a trivial feat. The problem is further complicated by the fact that the interpretation of experimental results for polycrystalline materials can be difficult, which makes computational techniques invaluable for providing insight at the atomic scale. Despite this fact, there is currently an alarming lack of atomistic computational studies of GBs and their effects in solid electrolytes, particularly regarding their impact on electronic structure. Some progress has been made regarding ion transport at solid electrolytes GBs using molecular dynamics studies with classical potentials. [10,[16][17][18][19][20] For example, it has been established why oxide-based solid electrolytes generally exhibit higher GB resistance [16] than sulfides and how GBs provide an explanation for the often observed differences in calculated and experimental Li-ion conductivities in some solid electrolyte families. [10] Although classical models are valuable as their low computational expense enables the investigation of large grain boundaries and nanostructures containing many tens or even hundreds of thousands of atoms over long timescales, their utility is limited by the fact that they lack access to electronic properties. Electronic properties are of increasing interest in the fields of solid electrolytes and solid-state batteries as it is these properties that underpin undesirable electrical conductivity and, especially at GBs, dendrite formation. [11,21]  In this study, we utilize first-principles calculations to elucidate how GBs impact the performance and properties of four archetypal solid electrolyte materials, namely, an oxide-based antiperovskite (Li 3 OCl) [22][23][24] and its hydrated counterpart (Li 2 OHCl), a thiophosphate ( -Li 3 PS 4 ) [8,25] and a halide (Li 3 InCl 6 ). [26,27] The GBs in all four materials are found to exhibit narrowed band gaps and charge-trapping states that would be conducive to unwanted electronic conductivity via polaron hopping and potential dendrite formation. By comparing the results of ab initio molecular dynamics (AIMD) simulations on the bulk and GBs of each material, it is found that the Li 3 OCl GBs pose large barriers to ionic conductivity whereas the GBs in Li 2 OHCl, Li 3 PS 4 and Li 3 InCl 6 are comparatively benign. While the contrast between GB resistance in oxides and sulfides is generally accepted, this study is the first to demonstrate the nature and origin of GB resistance in a halide-based solid electrolyte. We also show the significant influence that GBs have on correlated Li-ion transport for the first time. Our results provide design principles and important considerations for the synthesis and doping of solid electrolyte materials with high ionic and low electronic conductivity.

Grain Boundary Models
Stable GB structures are shown in Figure 1a-d and were determined using the methodology outlined in the Computational Methods section, which has been shown to be predictive of experimentally observed grain boundaries. [28][29][30] All GB models produced are stoichiometric and maintain the bulk phase composition. Furthermore, in the case of the polyanionic materials (Li 3 PS 4 and Li 2 OHCl) we ensured that the chosen terminations of the grains do not break any of the P−S or O−H bonds. Due to the large computational cost that would be incurred by very large grain boundary models, we aim to include no more than 400 atoms in our calculations while choosing tilt angles that allow for supercell lattice vectors of at least 10 Å parallel to the GB plane whilst maintaining grains around 15 Å thick.
The formation energies ( Figure 1e) for all of the considered GBs are low (0.25-0.63 Jm −2 ). The Li 3 OCl GBs have been modeled previously (with a mirror-symmetric geometry rather than with any optimized rigid-body translation as in this study) using classical potentials, [10] where they were also found to have low formation energies. There have been no previous atomistic studies of GBs in Li 2 OHCl, Li 3 PS 4 , or Li 3 InCl 6 , but the low formation energies indicate that our models are reasonable and representative. Such low formation energies also imply that similar GBs would be prevalent throughout the materials in equilibrium conditions, further highlighting the critical importance of the role they play in device performance.

Impact of Grain Boundaries on Electronic Structure and Conductivity
To determine the influence these GBs have on the electronic properties of the solid electrolytes, we calculated the projected density of states (PDOS) using the hybrid-DFT functional HSE06 (Figure 2). [31,32] The PDOS for the GB is obtained by projecting over only the region around the GB core, which is then compared with a PDOS obtained by projecting over the bulk-like region of the grain in the same supercell. Energies are referenced against the position of the valence band maximum (VBM) in the bulklike region of the cells. In general, the considered GBs exhibit reduced band gaps in the vicinity of the boundary (Figure 2e), with the changes in the Li 3 OCl and Li 2 OHCl Σ3{112} GBs being very small. In general, this corresponds to states appearing above the VBM, except for in the cases of the Li 3 OCl Σ5{310} and Li 3 InCl 6 Σ3{112} GBs, where there are significant contributions from states appearing below the conduction band minimum (CBM).
Narrowed band gaps may also imply that the GBs contain charge traps in their equilibrium geometry. In order to visualize the band edges, we add an electron and create a hole in each system without optimizing the geometry; the spin density isosurfaces associated with each case show where charge will prefer to localize. Evidence of electron trapping is not especially prevalent; whilst the Li 3 InCl 6 Σ3{112} GB shows some localization of electrons around In sites that form part of the edge-sharing octahedra, the electron density in the Li 3 OCl Σ5{310} is sufficiently diffuse that it does not appear in the isosurface plots. Conversely, evidence of hole trapping is far more pronounced and, in all cases where states appear above the VBM, there are highly localized holes centered on the anions in the vicinity of the GB, particularly on ions that are under-coordinated compared to the bulk.
The narrowing of band gaps at GBs represents a challenge in the development of solid electrolytes; an efficient ionic conductor must have poor electrical conductivity, which is achieved by having a wide band gap. Clearly, GBs with a very narrow band gap would have higher electrical conductivity which would encourage leakage current, reducing the efficiency of the device. For example, in the case where the narrowing of the band gap originates from states below the CBM, we would expect free electrons to segregate toward these lower-energy states at the boundary, leading to leakage current being dominated by the higher density of electrons in the GB. This has been shown to contribute to the formation of Li-metal dendrites or "filaments" along GBs in Li 7 La 3 Zr 2 O 1 2, [11] which would eventually lead to the failure of the device. None of the GBs studied here show severely narrowed gaps, and the localized nature of the GB states implies a high effective mass that would not be conducive to electrical conductivity. However, conduction of delocalized charge carriers within bands-as would be seen in a metal or a narrow-gap semiconductor-is not the only means by which electronic conduction takes place.
When an ion can exist in multiple charge states, it is possible that polaron formation can occur. This describes a process whereby a charge carrier induces a polarization in the lattice which, in turn, causes a charge carrier to become trapped on a specific ion as a polaron; the charge carrier has effectively trapped itself. Even though the charge carriers in polarons are highly localized, it is possible for polarons to hop between sites with relatively low barriers, [33][34][35] which may also contribute to an increase in electrical conductivity along GBs in these materials. In the case of Li 3 OCl and Li 3 PS 4 , the strong hole localization around the ions in the equilibrium geometry indicates that there may be some tendency towards forming polarons where a hole traps on either a O 2 − or S 2 − to form a O − or S − , respectively; evidence of polaron formation is reported across a wide variety of oxide and sulfide materials [36][37][38][39] and previous theoretical studies have shown that polaron trapping can be favorable in the vicinity of GBs. [28,40] Whilst the band edges in Li 2 OHCl are localized in a similar manner to Li 3 OCl, a polaron formed by trapping a hole on OH − to form a OH 0 is unlikely to be long-lived. Similarly, in Li 3 InCl 6 , hole polarons should not be expected to form through a hole trapping on a Cl − to form a Cl 0 , but it may be the case that electrons can trap on In 3 + to form In 2 + , which might also contribute to the formation of Li dendrites through the reaction In 2 + +Li + →In 3 + + Li 0 , for example.
In order to determine how large of a role polaron diffusion is likely to play in leakage current at GBs of solid electrolytes, we take the case study of a hole polaron localized at the Li 3 OCl Σ3{310} GB (Figure 3a) and a electron polaron in Li 3 InCl 6 Σ3{112} GB (Figure 3b). The ionization barrier of a polaron can be estimated by calculating the potential energy surface that interpolates between the polaron geometry and the equilibrium geometry corresponding to no trap. We find that neither polaron shows any barrier to trapping, indicating that the formation of these polarons is very favorable and that band-like conduction of their respective charge carriers would be unlikely at these boundaries. Polaron hopping rates can be calculated computationally using Marcus-Emin-Holstein-Austin-Mott (MEHAM) theory. [35,41,42] Essentially, all the information required to calculate polaron hopping rates from MEHAM theory can be obtained from the potential energy surface that interpolates between the geometry corresponding to the initial state and the geometry corresponding to the final state in which a polaron is localized on a neighboring site; a more complete description can be found in the Supporting Information. For the hole polaron in Li 3 OCl, we calculate an adiabatic activation energy of 0.30 eV and a 300 K hopping rate of 4.1 × 10 8 Hz. The electron polaron in Li 3 InCl 6 has a higher adiabatic activation energy of 0.45 eV and a correspondingly lower 300 K hopping rate of 6.4 × 10 5 Hz. Neither of these hopping rates is exceptionally high but, for the hole polaron especially, the rate is comparable that of some of the reported hopping pathways in TiO 2 which is generally considered to be a good polaronic conductor. [35,41,42] These results highlight important considerations that need to be made when synthesizing solid electrolyte materials. Even in the absence of extrinsic dopants, a material can have free charge carriers due to "self-doping" by intrinsic defects. For example, a recent study by Gorai et al., predicted that Na 3 PS 4 is intrinsically p-type due to self-doping by Na vacancies and that argyrodite Li 6 PS 5 Cl is intrinsically n-type due to self-doping by Li interstitials. [21] Then, by analysis of defect energetics, it is determined that higher synthesis temperatures would lead to higher intrinsic conductivity in the bulk of the material due to the higher concentration of the point defects responsible for self-doping. Let us consider the example of intrinsically p-type Na 3 PS 4 . It may be tempting to reduce synthesis temperatures as far as possible to suppress the formation of acceptor defects (to reduce hole concentration and associated conductivity) but lower synthesis temperatures would also lead to smaller grain sizes and increased numbers of GBs. Given the prevalence of states above the VBM across all the materials considered in our study (especially noticeable in the closely-related sulfide Li 3 PS 4 ) we should expect that these GBs would exhibit the levels of high hole conductivity that we would seek to avoid when choosing a low synthesis temperature. It is therefore clear that there is a fine balance between the controlling the relative proportions of point defects and extended defects. In comparison, the intrinsically n-type Li 6 PS 5 Cl poses a simpler problem; GBs with states above the VBM are less deleterious in an n-type material where there are fewer holes available to fill traps, so lower synthesis temperatures, leading to a larger number of GBs, would not be as much of a concern regarding electronic conductivity.

Ion Transport at the Microscale
A full understanding of solid electrolytes requires consideration of their ion dynamics. The activation energy, E a , and Li diffusivity for each GB system and the bulk of each material was calculated and is shown in Figure 4. Also shown are the relative activation energies in the GBs compared to bulk, ΔE a = E GB a − E Bulk a , decomposed into components parallel and perpendicular to the GB plane. The diffusion coefficients and activation energies are obtained from the entire simulation cell. Calculating diffusivity properties from only Li ions in the GB core region is possible; a previous computational study used such an approach and demonstrated that the decrease in diffusivity extends around 10 Å from the GB. [43] Given that our grain sizes are in the region of around 15-20 Å, we believe we are justified in the use of diffusion across the whole supercell as an approximation to the diffusion in the vicinity of the GB. Aside from the antiperovskite Σ5{310} GBs, there is not a significant difference in the activation energies through and along the GB planes. In Li 3 OCl (Figure 4a), we find that the presence of GBs has a profound effect on Li-ion conductivity, with values of E a increased by around 0.2 eV in both GB systems, which is in good agreement with previous calculations that employed classical interatomic potentials. [10] This provides further confirmation as to why computational predictions from simulations on bulk tend to overestimate the conductivity of antiperovskite materials compared to experiments which report activation energies as large as 0.6 eV, [23,44] which is in better agreement with our GB simulations than with our bulk simulations. For Li 2 OHCl, the increases in activation energy compared to the bulk are far less pronounced (Figure 4b). In the case of Σ5{310} GB, the activation energy through the GB plane is actually slightly lower than the bulk, though it is offset by an increased activation energy along the GB plane. Again, the systems containing GBs are closer to experimental reports of activation energies in the range 0.5-0.6 eV. [45,46] We also note that we do not observe any long-range proton diffusion in any of the Li 2 OHCl systems, which is in agreement with previous computational findings. [47] In comparison, for Li 3 PS 4 (Figure 4c) we find that the presence of GBs has a comparatively minor impact on ionic conduction, with activation energies for the bulk and the GBs ranging between 0.22 and 0.29 eV, which is in excellent agreement with experimentally reported values for crystalline -Li 3 PS 4 of 0.24 and 0.27 eV. [48,49] The lower impact of GBs in sulfides compared to oxides has been established previously using classical models and can be attributed to the stronger bonding between the cation and the oxide anion compared the sulfide anion. [16] The behavior of Li 3 InCl 6 GBs has not yet been reported, but we find that it is superior to the behavior of Li 3 PS 4 (Figure 4d), with small changes to E a of no more than around 0.04 eV. Our predicted activation energies of 0.30-0.34 eV for the GB systems agree remarkably well with reported experimental activation energies for Li 3 InCl 6 , which are in the range of around 0.32-0.35 eV. [26,27] Overall, the interfacial diffusion behavior of Li 3 InCl 6 shows an encouraging tolerance toward GBs. An obvious cause for the increase in the activation energy at a GB would be perturbations to the electrostatic potential in the vicinity of the boundary. Electrostatic perturbations may arise as a property intrinsic to the GB structure, or they may arise due to the segregation of charged defects leading to the development of a space charge region and a corresponding electrostatic barrier. [29,50,51] The electrostatic potential perturbations introduced by space charge regions can propagate into the bulk on the order of many nanometers (larger than our simulation supercells which are only around 3 nm long) and producing a quantitative continuum approximation to the space charge potential for all of the boundaries presented here would require a detailed analysis of defect energetics and segregation in each of the four materials and is beyond the scope of the current study. Nevertheless, examining the relative Li density as a function of distance from the boundary (normalized such that a density of 1 corresponds to the Li density in the bulk) gives us some indication of how Li segregates toward or away from the boundary (Figure 5, top panel of each plot) which can then be contrasted with observed electrostatic effects in the vicinity of the boundary. The total electrostatic potential experiences large variations due to alternating layers of cations and anions, as well as larger perturbations in the voids of the GBs which are not necessarily related to of the effects felt directly by the conducting Li ions. For this reason, we elect to focus on the average electrostatic potential in a sphere centered on each Li ion as a function of distance from the boundary, which we shall call ϕ Li , which illustrates of the electrostatic potential energy surface that the Li ions are moving in ( Figure 5, bottom panel of each plot). The procedure for these calculations is described in the Computational Methods section.
Both Li 3 PS 4 and Li 3 InCl 6 show a remarkably flat electrostatic profile, suggesting that the considered GBs do not have a strong impact on the electrostatic landscape of Li. Similarly, both Li 3 PS 4 and Li 3 InCl 6 show gentle variation in the density of Li across the supercell. This is not surprising, as defect concentrations are inextricably linked to electrostatic potential; a depletion of Li near the boundary implies an excess of negatively charged Li vacancies. The lack of sharp features in the electrostatic potential provides explanation for the small effect that they have on Li ion diffusion. In both of the Li 3 OCl GBs, however, we see depletion of Li ions in the vicinity of the boundary (to around 80% of the density in bulk) which is accompanied by large variations in the value of ϕ Li , with an absolute difference from the bulk value of around 0.20 and 0.10 eV for the Σ3{112} and Σ5{310} GBs, respectively, which aligns fairly well with the observed changes in E a . The Li 2 OHCl, in spite of its superficial similarities to the closely-related Li 3 OCl, shows a much weaker perturbation to ϕ Li in both boundaries. For the Σ3{112} this could be attributed to the lack of change in Li density across the boundary due to a lack of vacancy segregation but, for the Σ5{310}, the depletion in Li is nearly as severe as for the Li 3 OCl Σ5{310} albeit with the depleted region not propagating as far into the bulk, yet the perturbation to the electrostatic potential is far less prominent.
This stark difference in behavior between the two antiperovskites can, in part, be attributed to the mechanism of diffusion in these structures. The depletion of Li in the vicinity of the Σ3{112} corresponds to a negative electrostatic potential (Figure 5a), which implies that segregation of negatively-charged vacancies is responsible for this local perturbation to the electrostatic potential. If local structural properties reduce the ability of Li ions to move into the GB core region, then there will be an accumulation of Li in the layers adjacent to the GB core, which is visible as slightly higher Li density either side of the boundary in Figure 5a. Diffusion in stoichiometric Li 3 OCl is primarily "vacancy-driven," [52] meaning that Li ions diffuse by hopping into vacant sites while Li vacancies diffuse in the opposite direction. Such a mechanism is not problematic in bulk where vacancies are distributed homogeneously, but, if Li accumulates in a layer due to an extended defect, then there are fewer vacant sites in this layer and so fewer sites for Li ions to hop into. Such an issue was highlighted in a previous computational study of grain boundaries in lithium lanthanum zirconium oxide garnets; [43] many solid electrolytes are desirable because their bulk structures contain pathways that enable facile Li diffusion, but extended defects can disrupt these pathways. Diffusion in Li 2 OHCl is also vacancydriven, [53] but, due to the presence of protons, vacant Li sites are naturally incorporated into the structure, meaning that, even if Li struggles to move through the GB core, there will still be plenty of vacant sites for Li to diffuse through. This is reflected in the lack of Li accumulation either side of the Σ3{112} and so a lack of perturbation to the electrostatic potential (Figure 5c). The structure of Li 3 InCl 6 also contains vacant sites in the In layer through which Li is able to conduct, which may contribute to the small impact of GBs in this material.
This analysis does not fully explain the behavior of the Σ5{310} antiperovskite boundaries. Here, despite segregation of negatively-charge Li vacancies toward the boundary, we see a positive electrostatic potential. This suggests that the boundary is inducing a positive perturbation to the electrostatic potential which then encourages the segregation of negative Li vacancies. However, even significant Li vacancy segregation is not sufficient to negate this perturbation to the electrostatic potential in the Li 3 OCl (Figure 5b) and Li diffusion will be hampered by this remaining electrostatic perturbation. In the Li 2 OHCl, however, after a less pronounced segregation of Li vacancies, only small perturbations to the electrostatic potential remain (Figure 5d). This highlights the role played by the anions in electrostatic effects. A larger anion (such as S or Cl) will generally be more polarizable than a smaller one (such as O), and therefore will be able to more effectively screen electrostatic perturbations. The OH anion is similar in size to O, but differs by being a polyanion with a strong dipole which can undergo rotation and reorientation; such rotations would allow the polyanion to rotate to oppose electric fields created by electrostatic perturbations in the vicinity of the boundary, both reducing the drive for Li vacancies to segregate and reducing the long-range impact that local structural changes can have on the electrostatic potential. The reorientation of polyanions is often referred to as the "paddle-wheel effect." [54,55] The degree of polyanion reorientation occurring in a system can be quantified by calculating the vector reorientation autocorrelation function, C(t), which is defined as where u(t) is a unit vector from the center of mass of the polyanion (O in the case of OH) to a covalently-bonded atom (H in the case of OH). We observe much faster reorientation in the Σ5{310} GB of the Li 2 OHCl (Figure 6a), which we propose is due to anions rotating to oppose local perturbations in the electrostatic potential. This is supported by the Σ3{112} in which both the electrostatic environment and the degree of reorientation is more similar to the bulk. Polyanion reorientation is also known to occur in Li 3 PS 4 . The GBs in Li 3 PS 4 also show greater degrees of reorientation (Figure 6b). While this may act in tandem with the larger S ion to screen electric fields, the PS 4 polyanion does not have as strong a dipole as OH and so is likely to play a different role. The paddlewheel effect has been proposed to enable concerted motion. [54] By calculating the distinct part of the van Hove correlation function, G d (r, t), at 800 K for Li 3 OCl and 600 K for Li 3 PS 4 and Li 3 InCl 6 ( Figure S4, Supporting Information), we can determine a characteristic timescale of correlation in each of the systems, t c , which we define in this work as the lowest value of t for which G d (0, t) > 2. For context, using this metric, previous AIMD simulations carried out on a selection of "best-in-class" superionic conductors-namely, Li 10 GeP 2 S 12 , Li 7 La 3 Zr 2 O 12 , and Li 1.3 Al 0.3 Ti 1.7 (PO 4 ) 3 -yield values of around t c < 1.0 ps. [56] None of the materials considered in this study exhibit such highlycorrelated motion, but nonetheless provide a valuable case study.
The value of t c is calculated at 800 K for Li 3 OCl and at 600 K for Li 3 PS 4 , Li 2 OHCl and Li 3 InCl 6 . Transport in the bulk of Li 3 OCl does not exhibit strong correlation, with a large value of t c = 8.9 ps. For both the Σ3{112} and Σ5{310} GBs, the timescale of correlation is vastly increased to far beyond the 15 ps time window for which we have calculated G d (r, t) (see Figure S4, Supporting Information). Li 2 OHCl also shows little correlated motion, with neither the bulk nor either GB having a correlation timescale within the 15 ps window. In Li 3 PS 4 , we see much more strongly correlated motion in the bulk of the material with t c = 3.7 ps, with the Σ3{121} and Σ5{210} GBs having increased correlation timescales of around 7.1 and 11.7 ps, respectively. For bulk Li 3 InCl 6 , we find a higher correlation timescale of 6.3 ps, sitting roughly in the middle of of bulk Li 3 PS 4 and Li 3 OCl, but the presence of GBs has no real impact on t c , with the Σ3{112} and Σ5{310} GBs having timescales of around 6.1 and 6.8 ps, respectively (Figure 6c), which is perhaps not surprising given the lack of impact the GBs have on diffusivity at these temperatures.
Even if the degree of concerted motion in Li 3 PS 4 is not increased by faster reorientation, it might be expected that diffusivity along to the GB plane is faster than diffusivity through the GB, but we find that the difference in activation energy parallel and perpendicular to the GB plane is rather small (Figure 4c). Clearly, increased polyanion reorientation at the GBs is not increasing diffusivity beyond what would be seen in the bulk. Examining the radial distribution functions (RDF) provides some insight as to the function of PS 4 reorientation. The GBs in Li 3 PS 4 are less crystalline than the bulk, evidenced by the much broader and less-defined peaks in the P-P peaks (Supplementary Information), but despite this, the Li-S and Li-Li RDFs are strikingly similar to the bulk. We stated earlier that extended defects can disrupt diffusion pathways, but the extra degrees of freedom www.advancedsciencenews.com www.advenergymat.de provided by polyanions can act to remedy this effect by allowing Li to maintain a coordination environment similar to that of diffusion pathways in bulk, even in a disordered structure. Experimentally, samples of Li 3 PS 4 prior to calcining and sintering to produce a highly-crystalline material exhibit significant amorphous regions without significant loss of conductivity. [49] We propose that the good conductivity in low-crystallinity regions-whether that is in amorphous regions or in the vicinity of GBs-may be attributable to fast polyanion reorientation.

Conclusions
The structural, electronic, and ion transport properties of GBs are pivotal to the design of solid electrolytes for solid-state batteries. Despite this fact, our understanding of GBs in these materials is restricted by the current limitations facing both experimental and computational methods. In this study, we have investigated the significant effects that GBs have on four representative solid electrolyte materials (Li 3 OCl, Li 2 OHCl, Li 3 PS 4 , and Li 3 InCl 6 ) using first-principles simulations for the first time. The key results and associated design principles are summarized as follows: 1) It is found that for all four materials, GBs lead to reductions in the band gap, sometimes with the appearance of highlylocalized trap states for electrons or holes on ions which are coordinated differently than in the pristine bulk. Polarons formed at the boundary can hop with relatively low barriers, leading to increased electronic conductivity. Incorporating ions that are less likely to change charge state or tuning synthesis conditions to reduce the concentration of free carriers would mitigate this behavior.
2) The GBs in Li 3 OCl are found to lead to a severe reduction in Li-ion conductivity. In contrast, the GBs in Li 2 OHCl and Li 3 PS 4 are less severe, with Li 3 InCl 6 having particularly negligible effects on activation energy. This can, in part, be attributed to the GBs in these materials not significantly perturbing the electrostatic potential. Incorporating ions with higher polarizabilty or the ability to oppose electric fields by reorienting would mitigate the deleterious effects of electrostatic perturbations. 3) Extended defects can disrupt fast diffusion pathways found in the bulk of the material. In materials where diffusion is primarily "vacancy-driven" such as near-stoichiometric Li 3 OCl, segregation of defects to the boundary can block these diffusion pathways. Structures which inherently contain vacancies such as Li 2 OHCl or Li 3 InCl 6 are less prone to having pathways blocked in this manner. 4) GBs also lead to an increased degree of reorientation by polyanions. Incorporating polyanions such as PS 4 that are able to undergo reorientation can act to maintain bulk-like coordination environments even in low-crystallinity regions, enabling more facile diffusion.
By considering a range of materials simultaneously, these atomistic insights highlight a variety of important considerations that must be made when designing solid electrolyte materials. While this study is not exhaustive and focuses on highsymmetry boundaries, we emphasize that the main conclusions regarding anion chemistry and structure are not specific to high-symmetry boundaries and should still be expected to apply to lower-symmetry boundaries such as asymmetric tilt boundaries, twist boundaries, or arbitrary mixtures of tilts and twists.
We propose that future studies should focus more on detailed investigations into the specific mechanisms that have been hinted at here, for example charge trapping or complex polyanionic motion at GBs and other important interfaces. Augmenting our fundamental understanding of these interfaces and their complex influence is crucial for the future development of solid electrolytes for solid-state batteries.

Experimental Section
GBs were modeled in a periodic supercell containing two grains of finite width brought into contact to produce two symmetrically-equivalent GBs. It was ensured that the grains in each model were at least 15 Å thick in order to minimize interactions between the grains. To determine low-energy GB structures, a systematic scan was performed over all possible rigidbody translations between the grains, with the translation steps taken to be as close to 0.5 Å as possible whilst allowing the resulting grid of translations to be commensurate with the lattice vectors parallel to the GB plane. The structure with the lowest energy was determined to be a stable structure, where the formation energy, , in a periodic supercell is given by where E tot [GB] is the total energy of a GB supercell, N is the total number of units of bulk, E tot [Bulk] is the total energy of a unit of bulk, and A is the cross-sectional area of the slab where the factor of two accounts for the two equivalent interfaces in the model. This procedure had been used extensively in previous studies to successfully predict GB structures. [28][29][30]57] All geometry optimizations, AIMD, and electronic structure calculations were carried out using the implementation of DFT within CP2K. [58] Geometry optimization was performed until the force on ions was less than 0.01 eV Å −1 . When optimizing GB geometries, lattice vectors parallel to the GB were held fixed and the lattice vector normal to the GB was allowed to relax. The choice of exchange-correlation functional for standard calculations is PBEsol, [59] which generally performs well for solids. Double-basis sets optimized from molecular calculations (MOLOPT) [60] and Goedecker-Teter-Hutter pseudopotentials available within CP2K were used. [61][62][63] Five multigrids with a relative cutoff of 50 Ry and the finest grid having a cutoff of 500 Ry were used. CP2K was highly efficient when sampling only the Γ-point, so it was necessary to converge the size of the supercell in order to achieve proper reciprocal space sampling. For Li 3 OCl a supercell of 320 atoms was found to yield lattice parameters of a = b = c = 3.86 Å. For Li 3 PS 4 a supercell of 384 atoms was found to yield lattice parameters of of a = 6.11 Å, b = 8.01 Å, and a = 12.98 Å. For Li 3 InCl 6 a supercell of 360 atoms yielded lattice parameters of a = 6.45 Å, b = 11.05 Å, c = 6.36 Å, and = 111.1°. The parameters for each material were in good agreement with reported experimental results. [22,27,64] For the PDOS and partial charge density plots at each of the GBs, hybrid DFT functional HSE06 was used [31,32] to improve the band gaps and charge localization. The computational cost of hybrid calculations was reduced by the CP2K implementation of the auxiliary density matrix method (ADMM) [65,66] in which exchange integrals were approximated through mapping onto smaller, more localized basis sets.
All AIMD calculations were carried for at least 100 ps (with the exception of Li 2 OHCl, which used a smaller time step and ran for at least 80 ps) using the NVT ensemble with a Nosé-Hoover thermostat. Simulations were ran at temperatures of 600, 800, and 1000 K for Li 2 OHCl, Li 3 PS 4 , and Li 3 InCl 6 , and at elevated temperatures of 600, 800, 1000, and 1200 K for Li 3 OCl due to the low levels of diffusion observed at the GBs in this material. A timestep of 2 ps was used for the Li 3 OCl, Li 3 PS 4 , and Li 3 InCl 6 www.advancedsciencenews.com www.advenergymat.de and a timestep of 1 ps was used for the Li 2 OHCl. Self-diffusion data for the Li ions were calculated according to where ⟨r 2 i (t)⟩ is the mean squared displacement, D Li is the Li diffusion coefficient, and t is time. Mean squared displacements are calculated using the smoothing method implemented in Pymatgen's diffusion analysis modules. [67] These values can then be converted to conductivities, , using the Nernst-Einstein equation where n is the number density of Li, q is the charge of Li, k is the Boltzmann constant, and T is the temperature. Li ion vacancies were introduced into the systems at a concentration of 5% in order to encourage ionic mobility where the chargewas balanced by introducing a corresponding number of anion defects. Analysis of the trajectories for rotational dynamics was carried out using the TRAVIS code. [68,69] The normalized Li ion density analysis was performed by separating Li ions into bins of a width of 0.5 Åacross the entire trajectory and then dividing the number of Li in each bin by the volume of a 0.5 Å slice in the corresponding GB supercell. The point associated with each bin was smeared with a normalized Gaussian function with full-width half maximum of 0.5 Å to provide a continuous curve for visualization purposes. The smoothed density plot was produced by taking the mean Li-ion density between two peaks in this continuous curve. The average electrostatic potential around an Li ion, ϕ Li , was calculated by taking a sphere of radius 1.2 Å centered on the ion and then determining the mean electrostatic potential within this sphere. The electrostatic potential was obtained as volumetric data from CP2K.

Supporting Information
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