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Molecular Dynamics Studies on the Lithium Ion Conduction Behaviors Depending on Tilted Grain Boundaries with Various Symmetries in Garnet-Type Li7La3Zr2O12 Hiromasa Shiiba,† Nobuyuki Zettsu,*,†,‡ Miho Yamashita,† Hitoshi Onodera,† Randy Jalem,§,∥ Masanobu Nakayama,§,∥,⊥ and Katsuya Teshima*,†,‡

J. Phys. Chem. C 2018.122:21755-21762. Downloaded from pubs.acs.org by DURHAM UNIV on 10/01/18. For personal use only.

†

Department of Materials Chemistry, Faculty of Engineering and ‡Center for Energy and Environmental Science, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan § Global Research Center for Energy Based Nanomaterials Science (GREEN) and ∥“Materials Research by Information Integration Initiative (Mi2i)”, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan ⊥ Frontier Research Institute for Materials Science (FRIMS), Nagoya Institute of Technology, Gokiso, Showa, Nagoya, Aichi 466-8555, Japan S Supporting Information *

ABSTRACT: Grain boundary (GB) structure is a critical parameter that significantly affects the macroscopic properties of materials; however, the evaluation of GB characteristics by modern analytical methods remains an extremely challenging task. In this work, Li+ conductivity degradation at the GBs of cubic Li7La3Zr2O12 (LLZO) with a garnet framework (which represents the most promising candidate material for solid electrolytes utilized in all-solid-state batteries) has been investigated by various molecular dynamics approaches combined with newly developed analytical techniques. It was found that the transboundary diffusion of Li ions was generally slower than their diffusion in the bulk regardless of the GB symmetry; however, this effect strongly depended on the concentration of Li-deficient sites (trapping Li vacancies) in the GB layer. Furthermore, the compactness and density of the combined GB regions represent the key parameters affecting the overall Li+ conductivity of polycrystalline LLZO films.

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INTRODUCTION Grain boundaries (GBs) have been widely studied by various researchers because of their importance in scientific and technological applications. The addition of chemical dopants to GBs significantly affects the local structure and general material properties; for instance, it enhances the critical current density in high-temperature superconductors and facilitates oxygen incorporation into oxide-ion conductors.1−10 Therefore, various phenomenological understandings of the GBrelated chemical and physical processes have been achieved based on the results of both experimental and theoretical studies in thermodynamics.11 Although these findings can help to elucidate the general concepts of the GB-dependent phenomena, the atomistic understanding of movements inside GBs within a reasonable timescale has not been fully clarified, which severely limits our ability to control and design GB structures with specific properties for many technologically important materials. Recently, all-solid-state Li-ion batteries have attracted significant attention because of their high energy densities (originated from the device miniaturization) and high safety caused by their nonflammability.12,13 Many basic concepts of © 2018 American Chemical Society

all-solid-state (oxide) batteries are strongly related to various characteristics, such as safety and high energy density; however, the possible enhancements of their basic properties, which can be evaluated by the currently used analytical techniques, become complicated for various technical reasons. In particular, the Li+ conductivity at the interface and/or at the GB of oxide-based electrolytes is more than 3 orders of magnitude lower than those of other electrolyte systems, including liquids, gel-polymers, and sulfides, which represents the most important unresolved issue related to the full realization of all-solid-state batteries. Various oxide electrolytes such as perovskite-type Li3xLa2/3−x□1/3−2xTiO3 (LLTO), LISICON, NASICON, and garnet-type Li7−xLa3A2−x4+Bx5+O12 (A = Zr, B = Nb, Ta) have been widely investigated; in particular, garnet-type Li7La3Zr2O12 (LLZO) and its derivatives are considered promising materials because of their high Li+ conductivity of around 10−4 S cm−1 at room temperature [evaluated via Received: July 1, 2018 Revised: August 19, 2018 Published: August 23, 2018 21755

DOI: 10.1021/acs.jpcc.8b06275 J. Phys. Chem. C 2018, 122, 21755−21762

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Buckingham potential parameters, Aij, ρij, and Cij, were specific to the pairs of interacting species (the utilized simulation parameters are listed in Table S331). The DL POLY simulation package32 was used for all MD calculations (the corresponding time step was equal to 1 fs). First, the initial models were equilibrated for 20 000 time steps (20 ps) in the NPT ensemble at a temperature of 300 K. During this initial period, the volume of the cell was allowed to relax with time. The Nosé−Hoover thermostat and barostats33,34 were used to control the temperature and pressure, respectively. Afterward, the temperature was elevated from 300 to 1700 K at a rate of 10 K per 5 ps, and the NPT dynamic simulations were conducted for 100 ps, during which the cell angles were allowed to relax. After equilibration at 1700 K, the obtained structure was cooled to a desired temperature, which allowed the lattice to fully equilibrate in a shorter simulation time as compared to that required for the direct heating of the system to the target temperature. Cooling was performed via an iterative procedure, which involved decreasing the temperature in 10 K steps accompanied by dynamic simulations with durations of 5 ps. For each studied temperature, MD simulations containing 500 000 time steps (500 ps) were performed using the constant volume and temperature ensemble to obtain the statistical information about the diffusion rates. The bulk LLZO calculations were performed using the 3 × 3 × 3 unit cell superstructure, with the cubic symmetry containing 5184 atoms. The initial GB models with Li−Li, La−La, Zr−Zr, and O−O atomic distances shorter than 0.5 Å were eliminated to prevent unreasonable atomic repulsions. The electronic structure of the Σ3 (2−1−1) = (1−21) GBs was evaluated by conducting ab initio DFT calculations (the projected densities of states obtained for the La, Zr, and O atoms in the bulk and GB regions are shown in Figure S1). It was confirmed that the simulated GBs contained La3+, Zr4+, and O2− species. The GB energy γGB was defined as

electrochemical impedance spectroscopy (EIS)] and wide electrochemical window (they are neither oxidized nor reduced in a wide voltage range).14−18 Furthermore, computational studies have been performed to investigate the Li+ conduction behavior of these materials in the bulk state.19−21 Thus, Jalem et al. reported that Li+ migration is driven by the simultaneous cooperative motion characterized by long multiple-site successive hops with a very small timescale for fluctuations at intermediate positions.19 Despite the larger number of detailed studies utilizing various experimental and theoretical approaches,14−22 the experimentally determined Li+ conductivity did not precisely match the computationally predicted values. For example, the activation energy of cubic LLZO predicted via ab initio molecular dynamics (MD) simulations was equal to 0.1−0.3 eV,19−22 which corresponded to the Li+ conductivity of around 10−2 S cm−1 at 300 K. In contrast, the room-temperature activation energy of Li+ conduction in densely sintered ceramics manufactured from cubic LLZO (evaluated by EIS coupled with a numerical analysis approach using equivalent circuit models) was 0.29−0.4 eV (around 10−4 S cm−1).20,23−25 It may be difficult to argue about the importance of GB conductivity according to the impedance analysis based on the circuit models; the observed discrepancy can be attributed to the ambivalent properties of GBs. Various technological limitations of both the experimental and theoretical approaches utilized for elucidating the Li + conduction behavior of GBs (which originate from the low electron density of Li and low LLZO stability in the presence of the electron beam of a scanning transmission electron microscope) have restricted our ability of reaching a deeper understanding of the processes occurring at the atomic level. Furthermore, although the recent progress in computational science has considerably clarified the mechanisms of bulk and surface physicochemical processes, the investigation of the events occurring at GBs using ab initio density functional theory (DFT) remains an extremely challenging task because of the very large computational costs, which can be reduced by performing classical MD simulations.26 Very recently, Yu and Siegel demonstrated Monte Carlo simulations on the Li+ conduction behaviors at the GB in the LLZO solid electrolyte and indicated the possible presence of an anisotropic diffusion phenomenon, which appeared within the Σ3 boundary.27 In this study, we independently analyzed the Li+ conducting behavior of the LLZO cubic phase with tilted GBs with various symmetries by using classical MD approaches to examine their properties at the atomic level.

γGB =

1 (EGB − NE bulk ) 2A

(2)

where A is the GB area, EGB is the lattice energy of the GB model, Ebulk is the lattice energy per atom, and N is the number of atoms in a particular GB model. To investigate the ionic transport properties of a particular structure, the mean square displacement (MSD) of the ions was monitored as a function of time at different temperatures. For a system with N ions, the MSD of ion i at position ri(t + t0) and time t with respect to its initial position ri(t0) was defined as

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METHODS MD simulations were performed using a Born-like description of the ionic crystal lattice.28 The long-range Coulombic interactions were summed via the Ewald method,29 whereas the short-range interactions were described using the parameterized Buckingham pair potentials.30 The latter were summed to the cutoff value of 10.5 Å, beyond which the influence of the potential was considered negligible. The lattice energy was defined as ÄÅ É ij −r yz ij C yzÑÑÑÑ ÅÅÅ qiqj ij ij j z zz − jjj zzzÑÑÑ + Aij expjjj E L = ∑ ∑ ÅÅÅÅ jj ρ zzz jj r 6 zzÑÑÑ ÅÅ 4πε0rij ÅÇ i j>i Å (1) k ij { k ij {ÑÑÖ

N

⟨r 2(t )⟩ =

1 ∑ (ri(t + t0) − ri(t0))2 N i=0

(3)

The Li diffusion coefficient D was calculated from the MSD slope of the following function35 ⟨|ri(t + t0) − ri(t0)|2 ⟩ = 6Dt + B

(4)

where B is the atomic displacement parameter related to thermal vibrations. The Li ionic conductivity σLi was calculated using the Nernst−Einstein equation36 σLi = c Li(z LiF )2

where rij is the separation between the ions i and j; qi and qj are the ion charges; and ε0 is the permittivity of the free space. The 21756

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(5) DOI: 10.1021/acs.jpcc.8b06275 J. Phys. Chem. C 2018, 122, 21755−21762

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Figure 1. (a) GB energies calculated for each GB model at different temperatures and (b) its enlarged figure.

Figure 2. Correlations of the GB energies with the RDF differences between the bulk and GB models calculated for the (a) La−La, (b) Zr−Zr, and (c) O−O interactions.

where cLi is the Li carrier density, zLi is the Li charge, F is Faraday’s constant, R is the gas constant, and T is the temperature. The Li ionic conductivity values were calculated in the temperature range of 700−1700 K. As the obtained Li+ conductivities included both the bulk and GB components, their magnitudes were affected by the distances between the neighboring GBs. To eliminate the geometry factor, a special conversion formula was used to separate the Li ionic conductivities in the bulk and GB regions and estimate the local Li ionic conductivity at the GBs. It was assumed that the total resistance Rtotal utilized in the GB models contained two different types of resistances: bulk resistance (Rbulk) and GB resistance (RGB). In this case, the total resistance can be expressed as follows lR total = aR GB + (l − a)R bulk

atures using eq 2 for the evaluation of the GB contribution to the Li+ conductivity of LLZ, including Σ3 (2−1−1) = (1−21), Σ3 (100) × (2−12), Σ3 (1−10) = (0−11), Σ3 (110) × (411), Σ5 (031) = (03−1), Σ7 (3−2−1) = (2−31), Σ9 (1−14) = (−114), and Σ11 (1−13) = (−113). It was found that all the tilted GBs exhibited different formation energies. In particular, the tilted GBs represented by the Σ3 (2−1−1) = (1−21) and Σ3 (1−10) = (0−11) models were characterized by the first and second lowest GB energies of 0.42 and 0.74 J m−2 at 700 K, respectively, which were consistent with the thermodynamically stable faces of the LLZO crystal grown from a molten LiOH flux (see Figure S2). In contrast, the Σ3 (100) × (2− 12) GB model exhibited the highest GB energy of 2.06 J m−2 at 700 K. All the obtained GB formation energies were of the same order of magnitude, regardless of temperature. In addition, no significant changes in their values were observed below 1200 K, suggesting that the combined GB structures remained the same at the atomic level. However, their magnitudes significantly increased at temperatures greater than 1300 K, suggesting that the LLZ framework near the GBs was randomized via incongruent melting. Figures S3−S10 display the trajectories calculated with respect to the available La (light brown), Zr (gray), and O (red) crystallographic sites of the eight stoichiometric equilibrium atomic GB models by performing MD simulations at a temperature of 1300 K and duration of 500 ps. No migrations of La, Zr, and O atoms were observed. Each model consisted of the bulk and GB regions, with the disordered atomic arrangements at the center and both ends. The original garnet framework remained intact in the bulk region (without GBs) at temperatures below 1300 K. First, simulations using isothermal-isobaric (NPT) ensembles were performed for both the tetragonal and cubic frameworks as initial structures at a duration of 500 ps to determine the most energetically stable

(6)

where l is the distance between the GBs and a is the thickness of the GB region. The bulk resistance was obtained from the corresponding bulk calculations, and the thickness of the GB region was estimated from the variations of the Li + concentration along the axis perpendicular to the GB surface. After that, the RGB values were obtained for all the used GB models. The Li-ion conductivities derived from the values of RGB were considered the local conductivities within the GB regions. The fraction of the GB contribution to the total resistance x was computed using the following formula x=

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aR GB

aR GB + (l − a)R bulk

(7)

RESULTS AND DISCUSSION Figure 1 summarizes the GB formation energies calculated for eight stoichiometric equilibrium models at various temper21757

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Figure 3. (a) Three-dimensional average and local Li ionic conductivities calculated along the (b) a, (c) b, and (d) c axes using the bulk and GB models, respectively.

Figure 4. (a) Local Li ionic conductivity calculated along the c axis (across the GB layer). (b) Total and local Li ionic conductivities across the GB layers at 300 K obtained via MSD analysis.

bulk structure of stoichiometric LLZ. Their lattice energies and lattice constants as well as the Li occupancies of the 24d and 48g/96h sites were equilibrated at the same parameters of the cubic structure and temperature of 300 K after 20 ps, regardless of the initial configurations (see Figure S11 and Table S1). Thus, no criteria for distinguishing between the tetragonal and cubic phases were obtained in this study. Furthermore, the calculated lattice parameters were in good agreement with the experimental values (the detailed results are presented in the Supporting Information). Figure S12 displays the radial distribution function (RDF) plots obtained for the Li−Li, La−La, Zr−Zr, and O−O interactions in the bulk and at the GBs with different symmetries from the MD simulations at a total duration of 500 ps conducted at a temperature of 700 K (the differences between the bulk and GB models are summarized in Figure S13). Remarkable differences were observed for the La, Zr, and O sites, as compared to the Li atoms; in particular, the smallest values were obtained for the tilted GBs of the Σ3 (2−1−1) = (1−21) model and the largest ones for the GBs of the Σ3 (100) = (2−12) GB model. Furthermore, the atomic displacements within the GB regions correlated with the GB

formation energies. As shown in Figure 2, the observed trends were consistent with the GB energies of the used models, suggesting that the bulk LLZ framework remained virtually unchanged at the tilted GBs of Σ3 (2−1−1) = (1−21) with the smallest difference. In contrast, the atomic arrangements at the Σ3 (100) × (2−12) GBs with the highest difference was highly randomized, resembling an amorphous phase with a thickness of around 1.8 nm. Figure 3a−d describes the computationally predicted apparent Li+ conductivities of the bulk and GB models, including the average and separated conductivities along the a, b, and c axes (it should be noted that the Li+ conductivities of the utilized GB models included bulk conductivities because of the presence of bulk regions). The corresponding Li+ diffusion coefficient and Li+ conductivity at a specified temperature can be evaluated from the slopes of MSDs plotted against time and the Nernst−Einstein equation, respectively. Li ionic conductivity at room temperature was evaluated by the extrapolation of the results of high-temperature simulations because relatively long MD runs were required for conducting low-temperature simulations (below 700 K). It should be noted that at very short simulation times, the Li ions initially 21758

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Figure 5. Relationships between the GB energies and (a) local Li ionic conductivity across the GB layer and (b) total Li ionic conductivity along the c axis.

Figure 6. Changes in Li populations along the directions perpendicular to the GB layers calculated from the corresponding Li+ trajectories at 700 K using the (a) Σ3 (2−1−1) = (1−21), (b) Σ3 (100) × (2−12), (c) Σ3 (1−10) = (0−11), (d) Σ3 (110) × (411), (e) Σ5 (031) = (03−1), (f) Σ7 (3−2−1) = (2−31), (g) Σ9 (1−14) = (−114), and (h) Σ11 (1−13) = (−113) GB models.

located in the middle of the bulk region are unable to reach the GB region. According to the computational results presented in this work, the isotropic ion-conducting characteristics of bulk LLZO with a cubic lattice can be reproduced, as shown in Figure S14. In contrast, the results of the MSD analysis conducted for the tilted GBs indicate that the anisotropic Li+ conducting characteristics strongly depend on their symmetry. It was found that the bulk conductivities exhibited the highest values (in other words, the GB conductivity was smaller than the bulk one, regardless of the GB orientation), whereas the conductivities measured along the c axis (perpendicular to the GBs plane) were characterized by the lowest values. We found that the local conductivity across the GB plane primarily contributes to the reduction in the total ion conductivity of LLZO. It is consistent with the computational analysis for the Σ3 GB in LLZO, analyzed by Yu and Siegel using the static GB models.27 In particular, the local c axis conductivity within the GB layer is more than 1 order of magnitude lower than the total conductivity of the material (Figure 4a). Furthermore, the observed reduction in the Li-ion conductivity is highly dependent on the GB structure, which mirrors the trend observed for the activation energy of Li-ion diffusion (see Table S2). Although the majority of previously conducted EIS studies for the evaluation of Li-ion conductivity in the LLZO ceramics were unable to characterize the contribution of tilted GBs, the computations performed in this work strongly suggest

that the observed discrepancy between the experimental and computational Li+ conductivities can be potentially because of the effect of the GB resistances caused by their structural variations at the atomic level. The values of the fitted total average Li ionic conductivity (including both the bulk and GB components) and local Li ionic conductivity along the c axis (Li+ conductivity across the GB plane) calculated from the Arrhenius plots for all the GB models at 300 K are summarized in Figure 4b. The computationally predicted bulk Li+ conductivity was in good agreement with the data presented in previous reports;21,31 however, it was almost 2 orders of magnitude higher than the experimental value evaluated by EIS for the pelletized LLZO samples.17,25 The dependences of the local Li-ion conductivities (summarized in Figure 5a) on the GB formation energy (obtained for the energetically stable GBs) indicate that the local Li+ conductivity decreases with decreasing GB energy. It should be noted that the most energetically stable tilted GB of the Σ3 (2−1−1) = (1−21) model exhibited a relatively small conductivity, which was more than 3 orders of magnitude lower than the bulk value. As the total Li+ conductivity along the c axis demonstrated the trend observed for the GB formation energy (Figure 5b), it was concluded that the thermodynamically stable GBs were unable to provide a sufficiently fast diffusion of lithium ions in the cubic structure of the LLZO solid electrolyte (it should be noted that the 21759

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Figure 7. Contribution of the thickness of the GB layer to the total resistance as a function of the distance between the nearest-neighbor GB layers: (a) local conductivity along the c axis (across the GB layer) and (b) total three-dimensional conductivity.

fastest diffusion of Li+ was observed for the metastable GB structures of the Σ9 (1−14) = (−114) model). Calculating the time changes for the Li+ occupying the crystallographic sites of the studied region allows the prediction of possible ionic conduction routes in the LLZO framework at a specified temperature. The Li+ trajectories in the bulk LLZO depicted in Figures S15−S22 show that the Li+ species diffused along the 24d−48g/96h−24d continuous three-dimensional network pathways inside the bulk region, where each 24d tetrahedron site was connected to its neighbors via four face-sharing bridging 48g/96h octahedral sites. Such pathways are in good agreement with the recent results of the neutron powder diffraction analysis of LLZO37 and ab initio DFT and empirical computational studies.31,38 The crystallographic distortion resulting from the formation of tilted GBs deteriorated the Li+ diffusion properties. To further elucidate the Li+ diffusion characteristics at the GBs in terms of their chemical compositions, the variations of the Li+ concentrations near the GBs were calculated along the axis perpendicular to the GB surface (see Figure 6). The Li population was constantly modulated along the c axis in the bulk region; however, it was anomalously dispersed in the vicinity of the GB plane. For example, the average Li contents at the GBs were equal to 6.57 and 6.77 for the stable Σ3 (2− 1−1) = (1−21) and metastable Σ9 (1−14) = (−114) models, respectively. This reduction in the Li+ concentration suggests the formation of Li-deficient sites (trapping Li vacancies) in the GB region, which potentially represents the primary reason for the degraded ionic conductivity across the GB plane. Similar trends were obtained for all GB models utilized in our computational studies. The decreased number of Li-deficient sites in the combined tilted GBs lowered their Li-ion conductivities. Despite the different crystal structures examined in our study, a similar decrease in the Li ionic conductivity at the GBs was observed for perovskite LLTO.39 Finally, the contribution of the thickness of the GB layer to the total resistance of LLZO was investigated. As its magnitude utilized in the computational models (around 5 nm) was significantly smaller than the experimentally observed LLZO domain sizes formed in sintered ceramics (in the micrometer range), the calculated contribution of GBs was highly overestimated (compared to the experimental data). Ohta et al. found experimentally that the contribution of the GB resistance to the total resistance of the pelletized LLZO ceramics was equal to 12%.17 To minimize the difference between the computational and experimental results, the contribution of the GB layer thickness to the total resistance of LLZO was examined as a function of the distance between the

nearest GB neighbors (Figure 7). As expected, the GB contribution to the total resistance decreased with increasing distance; however, it was also found for the first time that its value strongly depended on the symmetry of the GB structure. For example, at a neighboring GB distance of 1 μm, the Σ9 (1−14) = (−114) GB model exhibited the smallest contribution of 9.3%; however, its magnitude increased to the maximum of 91% for the tilted Σ3 (2−1−1) = (1−21) GB model. When the thickness of the GB layer increased to 80 μm, the thermodynamically stable Σ3 (2−1−1) = (1−21) GB exhibited a contribution of 11%, which was close to the experimental value. Therefore, the performed simulations for the GB contribution to Li+ conductivity can be used for predicting the experimental data. The same approach can be applied to evaluating the effect of the thickness of the GB layer on the total average resistance in all directions. Thus, the GB contribution to the degradation of Li+ conductivity was relatively small. For example, the contributions of the GB layers with thicknesses of 1 μm to the total resistance determined using the Σ9 (1−14) = (−114) and Σ3 (2−1− 1) = (1−21) GB models were 3.0 and 5.6%, respectively, which suggested that the diffusion of Li+ ions along the GB plane could proceed much faster than that across the GB plane. Further studies on the effects of dopants such as Al, Ta, and Nb on the Zr site, which are often used to stabilize the cubic structure of LLZO in the experiments, will be the subjects of later articles. Briefly, we have reported the analysis of Li+ conductivity at the Σ3 (2−1−1) = (1−21) GB models for Li5La3Nb2O12 based on a similar MD analogy,40 and we found that doping significantly affected the stability and Li + conductivity at the GBs. In the present study, the energetics, compositions, and Li ionic transport properties of eight symmetrically tilted GBs of garnet LLZO were theoretically examined at the atomic scale to achieve a better understanding of the GB-dependent phenomena. The results of classical MD simulations revealed that Li+ conductivity within and across the GB layer was generally reduced to the bulk conductivity; however, this effect was highly dependent on the off-stoichiometric Li-ion composition with different GB structures, as indicated by the obtained atomic trajectories. Thus, the presence of Li+ vacancies inhibited the diffusion across the GB plane, leading to a decrease in conductivity by 4 orders of magnitude with respect to the bulk conductivity. Furthermore the largest decrease in conductivity was observed for the symmetrically tilted GBs with the lowest formation energies, indicating that the latter parameter correlated with the thermal stability of GBs. Their high symmetry can be maintained by removing the 21760

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(3) Sato, Y.; Buban, J. P.; Mizoguchi, T.; Shibata, N.; Yodogawa, M.; Yamamoto, T.; Ikuhara, Y. Role of Pr segregation in acceptor-state formation at ZnO grain boundaries. Phys. Rev. Lett. 2006, 97, 106802. (4) Klie, R. F.; Buban, J. P.; Varela, M.; Franceschetti, A.; Jooss, C.; Zhu, Y.; Browning, N. D.; Pantelides, S. T.; Pennycook, S. J. Enhanced current transport at grain boundaries in high-Tc superconductors. Nature 2005, 435, 475−478. (5) Shibata, N.; Pennycook, S. J.; Gosnell, T. R.; Painter, G. S.; Shelton, W. A.; Becher, P. F. Observation of rare-earth segregation in silicon nitride ceramics at subnanometre dimensions. Nature 2004, 428, 730−733. (6) Guo, X.; Waser, R. Electrical properties of the grain boundaries of oxygen ion conductors: acceptor-doped zirconia and ceria. Prog. Mater. Sci. 2006, 51, 151−210. (7) Lu, K.; Lu, L.; Suresh, S. Strengthening materials by engineering coherent internal boundaries at the nanoscale. Science 2009, 324, 349−352. (8) Lee, W.; Jung, H. J.; Lee, M. H.; Kim, Y.-B.; Park, J. S.; Sinclair, R.; Prinz, F. B. Oxygen surface exchange at grain boundaries of oxide ion conductors. Adv. Funct. Mater. 2012, 22, 965−971. (9) Nie, J. F.; Zhu, Y. M.; Liu, J. Z.; Fang, X. Y. Periodic segregation of solute atoms in fully coherent twin boundaries. Science 2013, 340, 957−960. (10) Feng, B.; Yokoi, T.; Kumamoto, A.; Yoshiya, M.; Ikuhara, Y.; Shibata, N. Atomically ordered solute segregation behaviour in an oxide grain boundary. Nat. Commun. 2016, 7, 11079. (11) Sutton, A. P.; Balluffi, R. W. Interfaces in Crystalline Materials; Oxford University Press: Oxford, U.K., 1995. (12) Thangadurai, V.; Narayanan, S.; Pinzaru, D. Garnet-type solidstate fast Li ion conductors for Li batteries: critical review. Chem. Soc. Rev. 2014, 43, 4714−4727. (13) Takada, K.; Ohta, N.; Tateyama, Y. Recent Progress in Interfacial Nanoarchitectonics in Solid-State Batteries. J. Inorg. Organomet. Polym. 2015, 25, 205−213. (14) Huang, M.; Dumon, A.; Nan, C.-W. Effect of Si, In and Ge doping on high ionic conductivity of Li7La3Zr2O12. Electrochem. Commun. 2012, 21, 62−64. (15) Narayanan, S.; Epp, V.; Wilkening, M.; Thangadurai, V. Macroscopic and microscopic Li+ transport parameters in cubic garnet-type “Li6.5La2.5Ba0.5ZrTaO12” as probed by impedance spectroscopy and NMR. RSC Adv. 2012, 2, 2553−2561. (16) Murugan, R.; Ramakumar, S.; Janani, N. High conductive yttrium doped Li7La3Zr2O12 cubic lithium garnet. Electrochem. Commun. 2011, 13, 1373−1375. (17) Ohta, S.; Kobayashi, T.; Asaoka, T. High lithium ionic conductivity in the garnet-type oxide Li7−X La3(Zr2−X, NbX)O12 (X=0-2). J. Power Sources 2011, 196, 3342−3345. (18) Li, Y.; Han, J.-T.; Wang, C.-A.; Vogel, S. C.; Xie, H.; Xu, M.; Goodenough, J. B. Ionic distribution and conductivity in lithium garnet Li7La3Zr2O12. J. Power Sources 2012, 209, 278−281. (19) Jalem, R.; Yamamoto, Y.; Shiiba, H.; Nakayama, M.; Munakata, H.; Kasuga, T.; Kanamura, K. Concerted Migration Mechanism in the Li Ion Dynamics of Garnet-Type Li7La3Zr2O12. Chem. Mater. 2013, 25, 425−430. (20) Miara, L. J.; Ong, S. P.; Mo, Y.; Richards, W. D.; Park, Y.; Lee, J.-M.; Lee, H. S.; Ceder, G. Effect of Rb and Ta Doping on the Ionic Conductivity and Stability of the Garnet Li7+2x-y(La3-xRbx)(Zr2yTay)O12 (0 ≤ x ≤ 0.375, 0 ≤ y ≤ 1) Superionic Conductor: A First Principles Investigation. Chem. Mater. 2013, 25, 3048−3055. (21) Burbano, M.; Carlier, D.; Boucher, F.; Morgan, B. J.; Salanne, M. Sparse Cyclic Excitations Explain the Low Ionic Conductivity of Stoichiometric Li7La3Zr2O12. Phys. Rev. Lett. 2016, 116, 135901. (22) Meier, K.; Laino, T.; Curioni, A. Solid-State Electrolytes: Revealing the Mechanisms of Li-Ion Conduction in Tetragonal and Cubic LLZO by First-Principles Calculations. J. Phys. Chem. C 2014, 118, 6668−6679. (23) Lee, J.-M.; Kim, T. Y.; Baek, S.-W. Abstracts of Papers, 2012 MRS Fall Meeting & Exhibit, Boston, Nov 25−30, 2012; Materials Research Society: Pennsylvania, 2012; J13.02.

highly mobile lithium ions from the GB region via geometry relaxation for 500 ps. As the calculations performed in this work were restricted to the stoichiometric composition of the studied system, it is highly probable that excessive lithium ions are present at the interface between the bulk and GB regions. In other words, at the infinite relaxation time (corresponding to a grand canonical ensemble), all excessive lithium atoms would diffuse to the LLZO surface to form a Li2O layer. From the obtained results, it can be concluded that the deterioration of ion conductivity across the GB layer can be minimized by the presence of tilted GBs with a lower symmetry. To achieve this goal, the formation of metastable GB structures through a kinetically controlled reaction must be investigated under diffusion-limited conditions, such as liquid phase sintering with a lower solubility solvent.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b06275.

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Li occupancies in cubic and tetragonal LLZO; local Li ionic conductivity across the GB layer and activation energies; parameters of the Buckingham interionic potentials; scanning electron microscopy images of LLZO, trajectories of Li, La, Zr, and O atoms for various GB models; variations of the lattice energies and lattice constants of cubic and tetragonal LLZO; RDF differences between the bulk and GB models; Li ionic conductivity in bulk LLZO along each axis; and partial density of state of the GB model (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (N.Z.). *E-mail: [email protected] (K.T.). ORCID

Nobuyuki Zettsu: 0000-0003-2838-3165 Katsuya Teshima: 0000-0002-5784-5157 Author Contributions

H.S., N.Z., R.J., and M.N. contributed to the computational study of the LLZO crystals and GBs. H.S. and N.Z. contributed in the drafting of this paper. N.Z., M.-a.N., and K.T. contributed in making the concept and design of this study. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partially supported by JST-CREST. R.J. acknowledges JST-PRESTO. M.-a.N. and R.J. acknowledges NIMS-Mi2i. The crystal structure figures were drawn with VESTA41 and VMD.42

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DOI: 10.1021/acs.jpcc.8b06275 J. Phys. Chem. C 2018, 122, 21755−21762