Insights into the Lithium-Ion Conduction Mechanism of Garnet-Type

Aug 18, 2015 - These sites form a 3D network of conduction pathways in which each 24d tetrahedron is connected to its neighbors by four face-sharing b...
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Insights into the Lithium-Ion Conduction Mechanism of Garnet-Type Cubic Li5La3Ta2O12 by ab-Initio Calculations Randy Jalem,*,†,‡ Masanobu Nakayama,†,§,∥ William Manalastas, Jr.,⊥ John A. Kilner,# Robin W. Grimes,# Toshihiro Kasuga,∇ and Kiyoshi Kanamura○ †

Unit of Elements Strategy Initiative for Catalysts & Batteries (ESICB), Kyoto University, Katsura, Saikyo-ku, Kyoto 615-8520, Japan Department of Materials Science and Engineering, Nagoya Institute of Technology, Gokiso, Showa, Nagoya, Aichi 466-8555, Japan ∥ Japan Science and Technology Agency, PRESTO, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan ⊥ CIC Energigune, Parque Tecnologico, C/Albert Einstein 48, CP 01510 Minano, Alava, Spain # Department of Materials, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom ∇ Department of Frontier Materials, Nagoya Institute of Technology, Gokiso, Showa, Nagoya, Aichi 466-8555, Japan ○ Department of Applied Chemistry, Tokyo Metropolitan University, 1-1 minami Oosawa, Hachioji, Tokyo 192-0397, Japan §

S Supporting Information *

ABSTRACT: Garnet-type solid electrolytes are a class of materials that could potentially revolutionize Li-ion battery technology. In this work, ab-initio-based MD simulations have been performed to investigate the ion dynamics in pure garnet-type cubic Li5La3Ta2O12 (LLTaO) over the temperature range from 873 to 1773 K. A strong tendency for disorder in the Li sublattice was verified for LLTaO that explains the relative ease of stabilizing the reported cubic phase for this material. The Li+ conduction mechanism was determined to be facilitated by a cooperative hopping process characterized by long, multiple-site successive hops with a very small time scale for fluctuations at intermediate positions. A comparative study is also carried out between LLTaO and garnet-type Li7La3Zr2O12 (LLZrO), another candidate solid electrolyte.

1. INTRODUCTION There is a great need for lithium-ion batteries which do not incorporate liquid- or organic-based electrolytes. This is driving research and development of solid electrolytes,1−8 which could revolutionize battery technology; their promising properties include the potential to be nontoxic, ease of preparation, low cost, stability during operation, and enhanced safety.9−11 To date, the leading inorganic materials that come close to conventional electrolytes are the superionic conductors Li10GeP2S12 and Li10SnP2S12, with conductivity on the order of 10−3 S/cm at room temperature.7,8 However, their application has been limited due to the expense of Ge and the tendency for sulfur-based compositions to react with air and moisture.12 Hence, other compounds have been sought that can meet many of the demands of a practical solid electrolyte under actual working conditions. One family of oxides that is currently being explored is the garnet-type conductors such as cubic Li 7 La 3 Zr 2 O 12 (LLZrO) 213,1 4 and Li 5 La 3 Ta 2 O 12 (LLTaO),15−17 which have shown promise not only because of their conductivity (10−4 and 10−6 S/cm, respectively, at room temperature) but also due to their excellent electrochemical stability against molten metallic Li,1,2,15,16,18,19 which makes it possible to use a Li anode in a protected lithium electrode design to achieve a high energy density. However, © 2015 American Chemical Society

one major concern is that their conductivity falls short in comparison with the sulfur-based solid electrolytes and marketleading liquid/polymer-based electrolytes. To address this, over the last 10 years efforts have been made to optimize Li-based garnets through chemical or structural modifications. For example, some studies investigated aliovalent doping at the La site in the composition Li5+xMxLa3−xTa2O12 (M = Ca, Ba).1,20−22 Another study involving Zr site substitution with Ta was carried out in the range of 0.4 ≤ x ≤ 0.6 to yield the nominal chemical formula Li7−xLa3Zr2−xTaxO12; a conductivity of about 1.0 × 10−3 S/cm at x = 0.6 was measured.22 In yet another similar study, Nb doping at the Zr site was explored for the LLZrO system, resulting in an optimum conductivity of 8 × 10−4 S/cm.23 These substitutions do not block the Li pathway and increase vacancies, promoting disordering in the Li sublattice, which in turn leads to cubic phase stabilization. On the other hand, aliovalent doping that blocks the Li pathway is also reported with similar effect and improved conductivity.24−29 In all these reports, it is evident that the garnet framework is capable of accommodating different cation Received: May 28, 2015 Revised: August 12, 2015 Published: August 18, 2015 20783

DOI: 10.1021/acs.jpcc.5b05068 J. Phys. Chem. C 2015, 119, 20783−20791

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such as for Li3Nd3Te2O12, full occupancy of the Td site is suggested as the preferred configuration; this leads to a very low mobility with an activation energy (Ea) of 1.22 eV and a maximum observed conductivity of 1 × 10−5 S/cm at 600 °C.35 A significant conductivity is then observed when the occupancy shifts to a distribution of Li in the Td and Og,h cages. This is true especially for Li-rich compositions such as LLTaO and LLZrO in which some of the Td and Og,h cages are vacated and populated, respectively. It is not surprising then that optimization is anticipated by controlling Li content, such in the LLZrO−LLTaO series, and this was confirmed by the previously mentioned cubic phase Li7−xLa3Zr2−xTaxO12 in which one of the highest reported conductivities so far was achieved at an intermediate composition.20 Two crucial factors that affect conductivity appear to be associated with the restriction imposed by site-to-site interatomic separation between Td and Og,h cages and the Li content within the garnet structure. For LLZrO, the combination leads to a cooperative-type migration instead of a single hopping-dominated process; this has been demonstrated by previous ab-initio and classical molecular dynamics (MD) studies on LLZrO.36−38 Probing Li+, especially in situ, still remains a big challenge experimentally. For example, limitations in X-ray diffraction (primarily caused by poor scattering of Li) have made it an arduous task to fine tune atomic displacement parameters.39−42 Lately, neutron diffraction has been the method of choice for providing crucial insight and understanding about Li dynamics as evidenced in several works.17,43−46 To gain more knowledge of Li dynamics in garnet-based solid electrolytes, we investigated cubic LLTaO by using abinitio MD calculations. With more vacancies, LLTaO is easily stabilized in the cubic phase, unlike LLZrO in which a lowconductivity tetragonal phase is favored at room temperature due to an ordering effect. This could be an advantage in terms of ease in fabrication and property control for LLTaO, as compared to LLZrO, which requires doping for the cubic

sizes with only very minimal changes in its structure. Furthermore, the strategy taken by experimentalists appears to focus on modifying the Li population, Li distribution, and lattice parameter. Most of these recently reported improvements in garnet-type ionic conductors achieved conductivities on the order of 10−4 S/cm at room temperature.30−34 Li mobility in garnets is noted to be a complex process owing to the diversity in composition (and by extension Li content), accessibility of pathways, and vacancy configurations. In the garnet structure, two distinct sites are available for Li (see Figure 1), namely, the 24d tetrahedral (Td cage) site and the 6-

Figure 1. Perspective view of the cubic Li5La3Ta2O12 (LLTaO) garnet structure with space group Ia3̅d. The connectivity of Td and Og,h cages within the LLTaO framework is indicated by dashed lines. Li atoms/ occupancy, La atoms, and TaO6 octahedra are shown in green/white spheres, orange spheres, and violet-red sticks, respectively.

fold-coordinated 48g position (Og) and its distorted 4-foldcoordinated split site 96h (Oh) site (or collectively, Og,h cage). These sites form a 3D network of conduction pathways in which each 24d tetrahedron is connected to its neighbors by four face-sharing bridging octahedra. There are 9 potential sites per host formula unit which lead to an average of 7 Li occupied2 Li vacancy and 5 Li occupiedand 4 Li vacancy for LLZrO and LLTaO, respectively. In the Li-poor regime

Figure 2. (a) Energy variation vs volume (rescaled with respect to the calculated lowest energy configuration) taken from LLTaO models with different Li−Li vacancy arrangements. (b) Li−Li radial distribution function (RDF) plots accumulated after 30 ps MD runs for cubic LLTaO at different temperatures. (c) Characteristic site linkages forming a ring structure within the 3D Li pathway of the garnet structure. 20784

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The Journal of Physical Chemistry C stabilization. However, the conductivity of the former has been reported to be lower despite its intrinsic Li disordering. Furthermore, understanding how LLTaO and LLZrO differ fundamentally in the governing factors of their Li+ transport could provide better guidance for experimentalists when formulating optimization strategies in garnet-based solid electrolytes. Thus, in this work, we evaluate the bulk ionic conductivity, statistical occupancy of Li in the Td and Og,h cages, and ionic conduction mechanism of LLTaO and compare the results with that of LLZrO.

Figure 2b shows the Li−Li radial distribution function (RDF) plots for LLTaO averaged from the 30 ps NVT MD runs at different sampling temperatures. Figure 2c shows a portion of the actual Li pathway that resembles a ring structure and is a characteristic site linkage (i.e., no direct jumping between distinct Og,h cages and between distinct Td cages nor jumping via interstitial/non-Li sites); only two immediate Og,h neighbors for each Td cage are shown (instead of four Og,h neighbors). One obvious feature about the local structure of the system is the absence of any RDF peaks below 2 Å, due to the prevalent Coulombic Li−Li repulsion in this region (interactions A and B in Figure 2c). The first nearest neighbor (NN) shell is mainly related to occupied coordination sites for Li24d− Li96h and Li96h−Li96h pairs (interactions C and D, respectively). It is also evident that with increasing temperature the peak widths become broader as a consequence of the rapid motion of Li by thermal motion between adjacent coordination sites (characterized by peak positions). These peaks become sharper when temperature is reduced, and Li become more localized.36 The distribution is expected to be independent of temperature and may consist, to a considerable extent, of structural variation without a significant span of structural distortion (see Figure 2a). The intermediate NN shell shown by the 873 K peak between 3 and 4 Å is contributed by distinct Og,h Li−Li pairs (interaction E). 3.2. Mean Square Displacements. The typical time average MSD plots for the constituent atoms in LLTaO (at 1273 K) are shown in Figure 3a (additional data for the 2 × 1 ×

2. COMPUTATIONAL DETAILS First-principles calculations are carried out within the density functional theory (DFT) framework as implemented in Vienna Ab Initio Simulation Package (VASP),47,48 with the generalized gradient approximation (GGA-PBEsol)49,50 and a projectoraugmented wave (PAW) approach.51−53 The fractional coordinates of atoms are fully relaxed via the conjugate gradient (CG) algorithm, with a plane wave cutoff energy of 500 eV; convergence criteria are set to 10−4 and 10−3 eV for the allowed error in total energy and the break condition for the ionic step, respectively. To determine the reasonable lattice constant for the MD cell, 6 randomly generated Li configurations within a primitive cell (4 formula units) are optimized (T = 0 K) with a 2 × 2 × 2 kpoint mesh, spin polarized. For the NVT MD run, an 8 formula unit cubic cell is used with edges of more than 10 Å and an optimized lattice constant. The energy cutoff is reduced to 380 eV and k-point grid to 1 × 1 × 1 in order to manage the computational cost. An equilibration step with 1 fs step size is initially carried out using a Nosé thermostat54 at a constant temperature of 1273 K for 5 ps. MD runs with the same thermostat setting are then performed using equilibrated cells at temperatures of 873, 1073, 1273, 1473, and 1773 K for 30 ps with 1 fs per step. The coefficient of diffusion (D) for Li is calculated according to Einstein−Smoluchowski equation55 D = lim t →∞[(1/2dt )⟨[ r (⃗ t )]2 ⟩]

Figure 3. Time average mean square displacement MSD plots of (a) constituent atoms at 1273 K and (b) Li atoms at different temperatures.

(1)

where d is equal to the dimension of the lattice on which diffusion takes place and r(⃗ t) is the displacement of Li at time t. The value of D at a specific temperature can be estimated from the slope of the diffusive regime of the time average mean square displacements (MSD).56

1 cell is available in the Supporting Information). Results confirm that only Li atoms diffuse within the garnet framework. Also, only local vibrations about the crystallographic sites are observed for La, Ta, and O (i.e., no crystal melting) as evidenced by the flat MSD trends. In Figure 3b, the diffusive regime of the MSD plot is visible for the temperatures investigated (873−1773 K) and the time scale of the NVT MD runs (∼30 ps). This allows us to estimate the diffusion coefficient through linear fitting. 3.3. Lithium-Ion Dynamics. Figure 4a displays the Li trajectory within the LLTaO structure derived from 30 ps NVT MD sampling at 1273 K. As can be observed, the Li pathway in LLTaO assumes a continuous 3D network characterized by Td−Og,h site connectivity (see Figures 1 and 2c) and the absence of direct jumps between distinct Og,h cages via other interstitial sites. This type of path strongly resembles that of LLZrO as suggested in recent analyses of neutron powder diffraction data as well as in computational studies using abinitio DFT and empirical modeling.36−38,43,44,57,58 Figure 4b displays the Arrhenius plots for LLTaO and LLZrO from MD

3. RESULTS AND DISCUSSION 3.1. Cell Optimization and Li−Li Radial Distribution Function. Figure 2a displays the energy variation per formula unit vs cell volume derived from structural optimization of several Li/Li vacancy configurations of cubic LLTaO. The variation in the cell volume is only 0.3% (with cell energetics variation of 0.30 eV/formula unit or 14 meV/atom among samples), suggesting a very small structural distortion with respect to Li arrangement. The calculated lattice parameter with lowest energy among samples (12.77 Å) is already within 1% difference when compared with the experimental value (12.81 Å).16 With this level of agreement, the optimized lattice parameter value is then used in the actual NVT MD sampling run which in turn allows for the statistical evaluation of Li distribution in the Td and Og,h cages. A similar result was also noted in our previous ab-initio MD study for LLZrO.36 20785

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experiments, which could significantly affect the conductivity behavior. If Al is the dopant, more Li vacancies (Al3+ ↔ 3Li+) are expected to form and it is easy to suspect that conductivity could be changed. However, doping with Al or Ga has been found to result in just a minor reduction of the conductivity.62 Instead, the change in lattice constant with increasing doping level was experimentally determined to have a more significant influence on Li-ion dynamics.63 Al addition does not contribute to the stabilization of the cubic form of LLTaO since the number of available vacancies per nominal formula unit (i.e., 4 Li vacancies and 5 Li) is already expected to sufficiently minimize the interionic Coulomb interaction, increase the overall entropy, and reduce the free energy gain from Li ordering toward the tetragonal phase transition at low temperatures.64 Significant Li+−H+ exchange reaction is known in LLTaO, LLZrO, and other garnet materials once exposed to air and often leads to variation in conductivty.65−70 In a recent classical computational study made by Wang et al. for LLTaO, a thermodynamic factor (Γ) correction was applied to their simulation-based bulk conductivity, and this has led to an improved agreement with the experimental data.45 In the present ab-initio-based MD simulation though, we could not extract Γ due to our relatively smaller model cell size. Nevertheless, our results agree well with Wang et al’s45 Γuncorrected bulk conductivity trend (see Figure 4b). Moreover, in the present work, we focused on investigating the ionic diffusion of LLTaO and understanding how its transport process differs fundamentally with that of garnet LLZrO (another promising but well-studied solid electrolyte material). We would like to point out as well that we were able to readily perform direct property comparison between LLTaO and LLZrO because our approach does not require any fitted empirical parameters (which should be highly transferable across many material data sets) for the definition of interaction potentials. To gain information concerning the diffusion mechanism, it is necessary to first analyze the atomic distribution in the Li sublattice. Figure 5a shows the statistical occupancy with respect to the available Td and Og,h cages as calculated from the 1273 K 30 ps NVT MD trajectories. The distribution strongly resembles that of occupancy disorder over the Td and Og,h cages, that is, a mixture of coordination environments. The average occupancies are calculated to be 47.1% and 32.2% for

Figure 4. (a) Perspective view of the Li trajectory (in yellow) taken from a 30 ps NVT MD simulation run (postequilibration) at 1273 K. TaO6 octahedra and La atoms are shown in sticks and orange spheres, respectively. Li atoms are removed for clarity. Blue areas are sectional view of the trajectory on the cell sides. (b) Conductivities derived from (NVT) MD runs and experiments for garnet LLTaO and LLZrO.

calculations and experiments. The Nernst−Einstein equation was used to calculate bulk conductivity, σLi, as given by59 σLi = c Li(z LiF )2 (DLi /RT )

(2)

where cLi is the carrier density, zLi is the valency of Li, F is Faraday’s constant, R is the gas constant, and T is the temperature. For LLTaO, the calculated Ea is 0.26 eV, and the extrapolated bulk conductivity at room temperature is 1.63 × 10−3 S/cm. This is 3 orders of magnitude higher than the highest reported experimental value, which is on the order of 10−6 S/cm (Ea,expt = 0.56 eV).14,60 Computational imprecision can be ruled out since the settings employed for the MD run in here is able to reasonably reproduce the experimental bulk conductivity of LLZrO (10−4−10−3 S/cm, Ea = 0.30−0.34 eV)2,28 on a similar order of magnitude. The calculated Ea for LLZrO36 also agrees with a recent DFT-based study.60 It is therefore suggested that other factors could have led to the observed difference between the conductivity of LLTaO and LLZrO. The first possible factor is the existence of a transport process at high temperature that does not proceed at room temperature (phase transition), as hinted by the bend in the experimental total conductivity plot for LLTaO in Figure 4b (from ref 15). The higher number of Li vacancies in LLTaO could lead to a different diffusion behavior than the mechanism proposed for LLZrO.36,37 Fitting at the high-temperature region of the experimental conductivity plot of LLTaO, the extrapolation to room temperature results in a total conductivity of 1.5 × 10−4 S/cm, which is only an order of magnitude lower than the bulk conductivity determined in the present calculation. The second possible reason is the consequence of grain boundary effects that can lower total conductivity; resolving the contributions of the bulk and grain boundary over a wide range of temperature in LLTaO has been known to be problematic.15 By definition, total conductivity includes the effect of grain boundary resistance, while bulk conductivity describes only that of the bulk material. Inada et al. investigated Al-free Li7−xLa3Zr2−xTaxO12 pellets for (0 ≤ x ≤ 1.0), with relative densities in the range 89−92%, and found a total conductivity trend with a maximum at x = 0.5 for the cubic phase.61 This result is consistent with those reported by Li et al. on Ta-doped LLZrO with Al inclusion.22 We suggest that further investigation should be made for x > 1.0 as LLTaO is at the extreme terminal of the Zr−Ta series composition for garnet, at x = 2.0, and grain boundary conditions for Ta-rich compositions (1.0 ≤ x ≤ 2.0) could be different than those in Zr-rich compositions (0 ≤ x ≤ 1.0). The third reason could be the unintentional doping with Al3+ or H+ that is noted in

Figure 5. (a) Average Li occupancy with respect to tetrahedral Td and Og,h cages in LLTaO and LLZrO from a 1273 K MD run within a NVT ensemble (site cutoff = 1.2 Å). Occupancy for LLZrO is from our previous ab-initio MD study.36 The cage index number represents the labeling assignment for all available cages inside the garnet cell models: 1−24 for the 24 Td cages and 25−72 for the 48 Og,h cages. (b) Plot of electrostatic site potential difference Δϕ, and schematic representation of Li migration energy profile between a Td and an Og,h cage (inset). 20786

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The Journal of Physical Chemistry C the 24d sites (or Td cages) and Og,h cages (note, each octahedral cage has one 48g and two 96h sites), respectively. The weak site preference for either cages is consistent with the fact that LLTaO is easily stabilized in the cubic phase.60 This also matches the results of LLZrO, with average values at 59.9% of the 24d sites (or Td cages) and 85.0% of the Og,h cages,36 so that the additional 2 Li are mostly accommodated on the latter cage. Another parameter that would affect how Li ions diffuse is the electrostatic site potential difference between Td and Og,h cages or Δϕ = ϕTd − ϕOg,h. The values of ϕTd and ϕOg,h are derived from the long-range interaction of ions and assumes an averaged distribution for a given ion i in its cage; it determines the depth of the site potential well. ϕ is described by the equation ϕi =

∑ Zj/4π ϵ0lij j

(3)

where Zj is the valence (oxidation state, nominal) of the jth ion in the unit of the elementary charge, ϵ0 is the vacuum permittivity, and lij is the distance between ions i and j. For the purpose of calculation, the 48g site is used as the reference position for Li for the Og,h cage. In both LLTaO and LLZrO, the Td cage is electrostatically more favorable than the Og,h cage for a Li ion (ϕTd,LLTaO = −1.72 e/Å and ϕOg,h,LLTaO = −1.55 e/Å, ϕ T d ,LLZrO = −1.44 e/Å and ϕ O g,h ,LLZrO = −1.10 e/Å, respectively). However, a stronger short-range repulsion effect that predominates locally when the Li concentration increases could force Li atoms to assume an arrangement that reduces their interaction, causing some Li to move away from the Td cages. This is confirmed in LLZrO with 7/9 occupied Li sites per formula unit (i.e., a higher occupancy for the Og,h cages for the disordered case36) and ordering at room temperature. As stated above, ϕTd and ϕOh are related to the actual depth of the potential well for each cage. The implication of this is that the magnitude of the energy barrier related to the transition state (Ea) will differ when the site potential difference between two adjacent cages (Δϕ) is significantly larger than zero. This leads to a direction dependence for local Ea in which when the overall barrier height is considered, one direction (Tdto-Og,h jump or vice versa) will have an actual barrier of less than Ea (see Figure 5b, inset). In the case of LLTaO and LLZrO, the Og,h-to-Td jump direction has a barrier less than Ea. However, Δϕ in LLTaO (−0.17 e/Å) is determined to be less negative than in LLZrO (−0.34 e/Å), which implies that the potential along the Td−Og,h−Td−... diffusion path for LLTaO has a less direction-dependent Ea. Aside from the lower Li content of LLTaO than LLZrO, leading to more vacancies, this smaller Δϕ could partly account for the high calculated bulk conductivity for LLTaO (based on the high-temperature MD data) as Li are less constrained in terms of which site to jump into around its adjacent coordination sites. Li-ion diffusion in LLTaO is represented in Figure 6a, which displays a series of snapshots taken from NVT MD sampling run at 1273 K (see corresponding animation video in the Supporting Information). Focusing on a local region such as the one depicted by specially colored atoms (at 20.00 ps), the hopping process starts with Li(2) jumping from its Td cage into the next Og,h cage (interaction C in Figure 2c) in the direction of Li(1), Li(3), and Li(4). This Li(2) jump is coupled with Li(1) and Li(4) jumps, with the latter two appearing to move out of the way of Li(2) by jumping 1 and 2 cages away, respectively, from their positions (at t = 20.50 ps). Li(2)

Figure 6. (a) Snapshot configurations of Li atoms from NVT MD run at 1273 K taken at 20.00, 20.50, 20.75, and 21.00 ps. Black and white spheres are the Td and Og cages, respectively. Green and specially colored spheres (1, aqua; 2, red; 3, blue; 4, violet) are Li. Td−Og,h vacancy pairs are indicated by asterisks (*). Animation video for a is available in the Supporting Information. (b) Displacement plots of selected (specially colored) Li in a. Shaded yellow area indicates concurrent jumps, and red dashed lines indicate the intersite (Td−Og) distance criterion (first peak in the RDF plot in Figure 2b). (c) Displacement plots at every 5 ps interval for (unique) Li atoms (each plot) undergoing multiple-site jumps.

continues to jump further by taking advantage of the Li divacancy near Li(1) (at t = 20.75 ps). Interestingly, Li(3) migrates as well (at t = 20.75 ps) by jumping toward the 20.50 ps position of Li(2). Finally, Li(2) makes a 2-cage jump before occupying an Og,h cage (at t = 21.00 ps) that is 5 cages away from its initial cage (when at t = 20.00 ps). The sequence of motions for Li(2) and Li(3) in which a cage formerly occupied by one Li is visited by another Li is a clear signature of a cooperative migration mechanism. Figure 6b presents the displacement profiles for Li(1)−(4) atoms, in which the displacement refers to the time-dependent migration distance, that is, the difference of position vectors, r, between simulation time t − t0 and t0 (t0 = 20.00 ps in here). As shown in the figure, the cooperative migration in Figure 6a proceeds at a very short interval (∼1 ps) as confirmed by the corresponding absolute displacement profiles in Figure 6b in which the average adjacent ionic site distance (the red dashed line) is crossed by Li(2); the time scale needed to overcome the intersite barriers here is much shorter than the time scale Li ions fluctuate about their sites. The presence of Li vacancy dimers (asterisk pairs in Figure 6a) seems to facilitate this transport behavior. On average, ∼10% of the Li atoms are confirmed to exhibit a migration behavior as observed for Li(2), over a 30 ps period (see Figure 6c). Over the same time scale, only a well-defined sequence of stepped single-site jump events is observed in LLZrO.36 The correlated hopping process can be further analyzed by investigating the temporal evolution of the distinct part of van Hove correlation function71 20787

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> rTd−Og,h, t). Meanwhile, a rapid contribution to Gd(r,⃗ t) is observed for a concerted motion (Figure 7d); one Li (j) contributes rapidly to Gd(0, t) (by filling Vf.o.,Og,h(i) in a very short time interval), and another (i) concurrently contributes rapidly to Gd(2rTd−Og,h, t) (relative to Vf.o.,Td(j)). Note that these descriptions only consider local regions, and the final Gd(r,⃗ t) over the relevant range of r is an averaged quantity. Also, for a system which is dominated by a single hopping process, correlation hole filling at r = 0 (with respect to Gd(r,⃗ 0), the static RDF) within short relaxation times should be significantly sluggish. Figure 8a−d shows the van Hove distinct part Gd(r,⃗ t) profiles at different MD temperatures for LLTaO. For comparison, we also calculated Gd(r,⃗ t) for LLZrO, and the results (at 1273 K) are shown in Figure 8e. As with RDF plots, thermal effects with increasing temperature cause the Gd(r,⃗ t) peaks to broaden as a result of increased vibration about the Td and Og,h cages as well as increase hopping to adjacent sites. For both LLTaO and LLZrO, the correlation related to the cooperative hopping is confirmed by the rise in Gd(r,⃗ t) around the r < 2 Å region according to Figure 8a−d and 8e, respectively; this corresponds to transitioning Li pairs for Td → Oh (j, 96h in octahedron 1) → Oh (i, 96h′ in octahedron 1) → Td′ and Oh (j, 96h in octahedron 1) → Oh(96h′ in octahedron 1) → Td(i) → Oh′(96h in octahedron 2). Meanwhile, the peak decrease for both materials for 2 Å < r < 3 Å can be assigned to the various rearrangements associated with Td (i)−Oh (96h in octahedron 1)−Oh (j, 96h′ in octahedron 1)−Td′ and Td(j)− Oh (96h in octahedron 1)−Oh (i, 96h′ in octahedron 1)−Td′ chains (interaction C in Figure 2c);72 this decrease is coupled to the probability build up with time for r < 2 Å. However, the local dynamics for 1 Å < r < 2 Å (interaction B in Figure 2c) hints at some differences in the contribution to the hopping processes between LLTaO and LLZrO within the same investigated time scale. The intermediate-range Li ordering in LLTaO also appears to gradually change as evidenced by the slight left shift in the position of peaks and valleys for 4.5 Å < r < 5.5 Å. This feature is obvious at 873 K (Figure 7a) and persists even at 1773 K (Figure 7d). However, for LLZrO (Figure 7e), no evidence is found for any change in the intermediate range ordering; this is consistent with previous calculations.38 On the basis of the evolution of the maximum Gd(r,⃗ t) value for r ≈ 0 (Figure 8f), the probability of finding a Li at later times at a site previously occupied by another Li is, on the average, 1.45 times higher in LLTaO than in LLZrO. This means that, interestingly, even with a lower number of neighbors statistically surrounding a vacated site in the first coordination sphere in LLTaO, hopping toward the aforementioned vacant site is relatively less restricted as compared to LLZrO. This mobility is realized by the successive jumps observed for several Li, moving via the vacancy dimers and only having a very low vibration time scale at intermediate position(s) (see Figure 6b and 6c). This causes the shift in Gd(r,⃗ t) at the intermediate range (4.5 Å < r < 5.5 Å) and consequently the additional superposed Td-Og,h intercage contribution for LLTaO (r < 2 Å, relative to LLZrO) in Figure 8a−d.

N

Gd( r ⃗ , t ) = (1/n)⟨∑ δ[ r ⃗ + rj⃗(0) − ri (⃗ t )]⟩ i≠j

(4)

where N is the number of Li, δ[·] is the three-dimensional Dirac delta function, while rj⃗ and ri⃗ are displacements of particles j and i, respectively, at time t. This function describes the probability density of finding particle j at location r ⃗ after time t, in relation to the position of another particle i at the initial time t = 0. When t = 0, Gd(r,⃗ t) collapses to the more commonly known static RDF plot. Another feature of the Gd(r,⃗ t) graph is that when rj⃗ (t) − ri⃗ (t = 0) goes to zero, the replacement of atom i at its site by atom j takes place. As the frequency of this replacement event increases, the probability builds up rapidly for Gd(r,⃗ t) near r = 0, which is an indication of cooperative hopping within the geometric confinement of the 3D Li pathway (see Figure 4a). Upon closer inspection, this cooperative motion signature for atom i and atom j is strongly linked to the vacancy site which was formerly occupied by either one, Vf.o.. The analysis can be further aided by considering a pathway segment formed by the crystallographic site linkages: Td for the 24d site, Og,h octahedra with two 96h sites and one Og for 48g site in octahedron 1, Og,h′ for octahedron 2, and so on. Li atoms i and j occupying any of the tetrahedral and octahedral cages in a path segment can be labeled as Td(i), Td(j), Og,h(i), Og,h′(i), Og,h(j), Og,h′(j), etc. Figure 7 illustrates some of the basic dynamics and relation to

Figure 7. Idealized representation of some basic ion dynamics along the Li pathway and its relation toward contributing to the distinct part Gd(r,⃗ t) of the van Hove function: (a) Li j (green circle) jumping to a neighboring cage previously occupied by Li i denoted as Vf.o. (contributes to Gd(0, t) by filling the correlation Vf.o.,Td(i) (hatched circle)), (b) Li j jumping to a neighboring cage that is next to Vf.o.,Td(i) (contributes to Gd(rTd−Og,h, t)), (c) Li i in a back and forth jump, leaving Vf.o.,Td(i) (main contribution to Gd(r > rTd−Og,h, t) with respect to other Li excluding i), and (d) Li i and j jumping in a concerted manner (Li j contributes to Gd(0, t) by filling Vf.o.,Og,h(i), and Li i contributes to Gd(2rTd−Og,h, t) relative to Vf.o.,Td(j)).

the Gd(r,⃗ t) contribution within a given time interval. When a Li (j) jumps to a neighboring cage previously occupied by another Li (i) (Figure 7a), a formerly occupied cage for r = 0 (Vf.o.,Td(i)) is filled, and this increases Gd(0, t). A similar case is noted when jumping to a neighboring cage that is next to a cage previously occupied by another (i) (Vf.o.,Td(i)) (Figure 7b); this increases Gd(rTd−Og,h, t). For a back and forth jumping (Figure 7c), Vf.o.,Td(i) is generated and will be referred to by other Li for Gd(r

4. CONCLUSIONS Li-ion dynamics in garnet-type LLTaO was successfully investigated using constant NVT MD simulations. We confirmed by site occupancy analysis the strong tendency for Li disordering in LLTaO which explains the relative ease in 20788

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Figure 8. Distinct part Gd(r⃗, t) of the van Hove correlation function for Li at (a−d) different temperatures (873, 1273, 1473, and 1773 K) for LLTaO and (e) 1273 K for LLZrO, and (f) evolution of the maximum Gd(r⃗, t) (at 1273 K sampling) for r ≈ 0.

Notes

stabilizing its cubic symmetry. There is a relatively less direction-dependent Ea along the diffusion path in LLTaO than in LLZrO, which can be explained by the former’s lower electrostatic site potential difference (i.e., between Td and Og,h). This characteristic, coupled with the presence of vacancy dimers, leads to the unique cooperative hopping mechanism in LLTaO, characterized with long, multiple-site successive jump events with a very small time scale for fluctuations at intermediate positions. In LLZrO, ion transport also follows a cooperative mechanism but is mainly limited to single-site successive hopping.

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The present work was partially supported by JST, PRESTOprogram and MEXT program “Elements Strategy Initiative to Form Core Research Center” (Since 2012), MEXT; Ministry of Education Culture, Sports, Science and Technology, Japan. M. N. thanks Nagoya Institute of Technology for the financial support. Crystal structures were drawn with VESTA.73 Animation video was created using the VMD software.74

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ASSOCIATED CONTENT

S Supporting Information *

(1) Thangadurai, V.; Weppner, W. Li6ALa2Ta2O12 (A = Sr, Ba): Novel Garnet-Like Oxides for Fast Lithium Ion Conduction. Adv. Funct. Mater. 2005, 15, 107−112. (2) Murugan, R.; Thangadurai, V.; Weppner, W. Fast Lithium Ion Conduction in Garnet-Type Li7La3Zr2O12. Angew. Chem., Int. Ed. 2007, 46, 7778−7781. (3) Inaguma, Y.; Liquan, C.; Itoh, M.; Nakamura, T.; Uchida, T.; Ikuta, H.; Wakihara, M. High Ionic Conductivity in Lithium Lanthanum Titanate. Solid State Commun. 1993, 86, 689−693. (4) Alpen, U. V.; Schulz, H.; Talat, G. H.; Böhm, H. OneDimensional Cooperative Li-Diffusion in β-Eucryptite. Solid State Commun. 1977, 23, 911−914. (5) Hong, H. Y.-P. Crystal Structure and Ionic Conductivity of Li14Zn(GeO4)4 and Other New Li+ Superionic Conductors. Mater. Res. Bull. 1978, 13, 117−124. (6) Goodenough, J. B.; Hong, H. Y.-P.; Kafalas, J. A. Fast Na+-Ion Transport in Skeleton Structures. Mater. Res. Bull. 1976, 11, 203−220. (7) Kamaya, N.; Homma, K.; Yamakawa, Y.; Hirayama, M.; Kanno, R.; Yonemura, M.; Kamimaya, T.; Kato, Y.; Hama, S.; Kawamoto, K.;

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b05068. Comparison of the MSD trend between 1 × 1 × 1 and 2 × 1 × 1 Li5La3Ta2O12 molecular dynamics cell under NVT ensemble condition at 1273 K (PDF) Animation video of Li atoms related to Figure 6a (MPG)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +81-29-860-4953. Fax: +81-29-860-4981. Present Address ‡

R.J.: National Institute for Materials Science (NIMS), Global Research Center for Environment and Energy based on Nanomaterials Science (GREEN), Namiki 1-1, Tsukuba, Ibaraki, Japan, 305-0044. 20789

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