Origin of Fast Ion Conduction in Li - American Chemical Society

Dec 5, 2016 - fractional occupancy for Li sites leads to different c channel ... novel mechanism of correlated diffusion between the c channel and the...
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Origin of Fast Ion Conduction in Li10GeP2S12, a Superionic Conductor Arihant Bhandari and Jishnu Bhattacharya* Department of Mechanical Engineering, IIT Kanpur, Kanpur, Uttar Pradesh 208016, India ABSTRACT: The discovery of additional sites has revealed important insights into the possibility of cross-channel ionic diffusion in Li10GeP2S12 (LGPS). Here, we present an ab initio study of a Li diffusion mechanism in LGPS. The presence of fractional occupancy for Li sites leads to different c channel population of Li and different local environment for Li hop both along the c-directional channels and in the ab plane. We perform a rigorous computation of the energy barriers with different population of Li and different environment of Li surrounding the c channels. It is found that the diffusion energy barrier in LGPS depends significantly on channel population, local environment, and the type of hop which leads to a range of relative diffusivity along different directions. A novel mechanism of correlated diffusion between the c channel and the ab plane is found with significantly low energy barrier. The relative frequency of occurrence of the novel hop mechanism critically controls the degree of anisotropy in LGPS. Moreover, low activation barrier associated with this hop leads to high diffusivity which is sustained even after repeated cycling due to the presence of cross-channel pathways.

1. INTRODUCTION Li-ion batteries have important applications in renewable energy storage, electric vehicles, portable electronics, space exploration, and defense technology. The discovery of solid electrolytes, which have high conductivity and wide potential window, has made possible the implementation of all-solid-state batteries which are envisioned as compact, safe, and electrochemically more stable.1 There are two major families of compounds which are currently being explored for the solid electrolyte application: garnet type (Li7La3Zr2O12, Li5La3Ta2O12, etc.) and LGPS type (Li10GeP2S12, Li2S−P2S 5, Li3N, Li3PS4, Li3PO4, Li7PS6, Li6PS5Cl, Li4P2S6, Li4P2O6, Li4P2S7, Li4P2O7, etc.). Among all these compounds, the highest room-temperature conductivity (12 mS/cm) has been reported for Li10GeP2S12 (LGPS).2 We attempt to understand the mechanism of fast ion diffusion in this compound in the present study. The phenomenal conductivity and the crystal structure of LGPS were reported by Kamaya et al. in 2011.2 LGPS has a tetragonal structure with lattice constants a = 8.71 Å and c = 12.63 Å. The crystal structure belongs to the space group P42/ nmc. Kamaya et al. found that the diffusion is anisotropic where the c directional motion dominates over the ab plane. The discovery was followed by an ab initio molecular dynamics study by Mo et al., who found an average energy barrier of 0.17 eV along the c channel and 0.28 eV in cross-channel direction, i.e., along the ab plane.3 Further, Xu et al. found that the Li atoms move along the c channels in a cooperative way, instead of through the nearest neighbor hops, due to the large Coulombic repulsion between them.4 Adams et al. performed a molecular dynamics study and predicted the presence of © 2016 American Chemical Society

additional Li sites which would provide diffusion along the ab plane.5 The additional sites could change the diffusion scenario altogether by not only changing the occupancies of Li in the c channel but also providing a diffusion mechanism involving the ab plane. Later, Kuhn et al. confirmed the presence of additional sites experimentally by performing single-crystal XRD.6 The interesting discovery of additional sites opens up the possibility of hitherto unknown atomic mechanisms of ion movement. Kuhn et al. in a further study reported that the diffusion in LGPS is more or less isotropic with an overall activation energy barrier of 0.22 eV.7 Although the computational studies on LGPS, prior to the discovery of additional sites, gave important insights about the diffusion mechanism, their results need to be reconsidered as the additional sites would change not only the atomic arrangements but also the kinetics. Du et al. studied the LGPS material with DFT tools and reported a c-directional energy barrier of 0.23 eV and an ab plane energy barrier of 0.37 eV.8 Both the numbers were higher than the previously reported values.3 However, the numbers verified the anisotropy and the diffusional preference along the c channel. Further, Hu et al. reported the ab initio calculations by including the van der Waals interactions and found a dramatically different result: c channel energy barrier of 0.23−0.27 eV and ab plane energy barrier of 0.11−0.18 eV.9 According to them, diffusion is dominant along the ab plane, and LGPS is a 3D ionic conductor. The view of Li-ion diffusion in LGPS as presented Received: November 2, 2016 Revised: December 5, 2016 Published: December 5, 2016 29002

DOI: 10.1021/acs.jpcc.6b10967 J. Phys. Chem. C 2016, 120, 29002−29010

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The Journal of Physical Chemistry C

The migration barriers for Li hops are calculated by using the nudged elastic band (NEB) method as implemented in VASP. Initially single atom Li hops were considered. We found that single Li atom hops are not feasible as any Li atom encounters significant repulsion from the next Li atom along the c channel and bounces back to its initial position upon relaxation. Hence, a collective motion of atoms along the c channel has been considered which is discussed in detail in section 3.3.1. A similar observation has been found by Du et al.8 and Xu et al.4 In the present study, we have also considered the effect of c channel population for collective motion along the c channel, which is discussed in detail in section 3.2.1. Along the ab plane as well, the isolated motion of the Li atom is not possible; hence, the collective motion of Li atoms at the additional 4c sites has been considered as discussed in section 3.3.2. A new mechanism of hop, viz., correlated hop, has been found, which contributes to diffusion along both the ab plane and c channel as discussed in section 3.3.3. Interactions between the periodic images may affect the energy barrier in the primitive unit cell. Therefore, we have tested the convergence of NEB calculation with respect to the cell size. NEB calculations have been performed for supercell of volume 2 × 2 × 1 for both the c channel and the ab plane hops. For the c channel hop, the energy barrier for the 1 × 1 × 1 primitive unit cell is 8% higher than that for the 2 × 2 × 1 cell, and for ab plane hop the energy barrier for the primitive unit cell is 9.6% lower than that for the 2 × 2 × 1 cell. As the changes are less than 10%, the primitive cell has been used for all NEB calculations in the present study. The remarkable match of the overall energy barrier with the experimental results (as mentioned in the Conclusions) indirectly justifies the adequacy of the method as well as the choice of the supercell. The formation energy (Ef) of a configuration is calculated as

by Hu et al. was not supported by any other study in the literature. Experimental solid-state NMR study by Liang et al. gives a c channel diffusion energy barrier of 0.16 eV and an ab plane energy barrier of 0.26 eV, which supports an anisotropic diffusion along the c channel.10 Clearly, there is a lot of variation in the degree of anisotropy in diffusion in LGPS as reported in the literature, especially between the experimental observations and the theoretical calculations. The key role of additional sites in the diffusion process is not fully understood. In the current paper, we perform a computational study on the diffusion process in LGPS using density functional theory. We include the additional sites in finding the ground state structure and the possible hop mechanisms in the atomic scale. The inclusion of the additional sites opens up the possibility of different hop mechanisms, and we have discovered a correlated hop with low activation barrier which contributes to both c channel and ab plane motions. The new hop mechanism is found to be the key in explaining the anomalously fast diffusion in this lithium superionic conductor.

2. METHODOLOGY All the calculations in the present study use density functional theory within the generalized gradient approximation (GGA) as implemented in the Vienna Ab initio Simulation Package (VASP).11,12 We have employed the PBE (Perdew−Burke− Ernzerhof) version of the projector augmented wave (PAW) pseudopotentials for the electronic core states.13 We use 500 eV for the energy cutoff for the plane-wave basis set and a 3 × 3 × 2 Monkhorst−Pack k-point mesh, which ensures energy convergence. A similar method and k-point mesh was also used in the DFT study by Du et al.8 The convergence criteria for energy and structural relaxation are set to be less than 1.0 × 10−5 eV and 0.01 eV/Å, respectively. The issue of finding the initial stable structure computationally is nontrivial. Out of the 32 possible Li sites in the primitive cell, 20 are occupied, making the total number of possible combinations as high as (32C20 = ) 225,792,840 (without taking into account the symmetry of the structure). Further, the fractional occupancies of Ge and P give rise to 3 possible orderings of Ge−P for the primitive cell, which makes the configurational space even larger. However, taking into account electrostatic interactions between Li placed at different sites, several relatively stable configurations can be found. Du et al. performed first-principles DFT-based calculations of several configurations and found that the ones with zigzag ordering of Ge−P are more stable.8 We have used the same zigzag arrangement of Ge−P to construct the ground-state structure of LGPS. The fractional occupancies as given by Kuhn et al.6 have been used to build the c channel population (explained in section 3.2.1) and local environment (explained in section 3.2.2) in LGPS structure. The primitive cell of LGPS has been used for all calculations in the present study. The same choice of cell size has been made in all the earlier studies on LGPS based on DFT.3,4,8,9,14 The restriction on the size arises majorly due to the exponential nature of the increase in configurational space with cell size. The configurational space within the primitive cell itself is huge (32C20, not accounting for the symmetry). For a supercell twice as big as the primitive cell, the configurational space becomes 64 C40 which is about 109 times bigger than that of the primitive cell.

Ef = E Li10GeP2S12 − 10E Li − EGe − 2E P − 12Es

(1)

where the terms on the right-hand side are the total DFT energies at 0 K for LGPS, Li, Ge, P, and S, respectively. We calculate the migration energy barrier for Li hop along the c channel for all possible environments of neighboring Li atoms surrounding the c channel. If N is the number of all possible configurations (including the degenerate ones), the average energy barrier ( E b) can be calculated as Ef

Eb =

( )

∑i ∈ N E bi exp − kTi Ef j

( )

∑j ∈ N exp − kT

(2)

where k is the Boltzmann constant; T is the absolute temperature; Efi is the formation energy of the ith configuration; and Ebi is the associated energy barrier for Li hop originating from that configuration. In the calculations of the present study, absolute temperature is 298 K, i.e., the standard room temperature. The value of E fi decides the contribution of the i th configuration in the overall average energy barrier ( E b). A particular bunch of lower formation energy configurations will have the maximum contribution in the average. If N0 is a subset of N, such that the members of N0 have formation energy significantly lower than the members of subset N − N0, we can exclude the configurations having higher formation energies 29003

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Figure 1. Presence of additional sites: (a) the Li atom at the 16h site in a densely populated c channel is found to relax to the vacant additional 4c site and (b) the Li atom at the additional 4c site is found to relax to the vacant 16h site in a sparsely populated c channel.

Figure 2. Liquidity of Li sublattice: (a) the distances (in Å) between the Li atoms in the front c channels (dark green Li atoms) in the unrelaxed configuration and (b) the distances (in Å) in the relaxed configuration.

occupation and migration are included. Therefore, our primary task is to check whether DFT finds the proposed additional sites by the aforementioned authors. Toward this purpose, we relax the structure as proposed by Kamaya et al. for various configurations of Li and find that Li atoms in some configurations move to the additional sites (4c) as found by Kuhn et al. One of such configurations is shown in Figure 1(a), which shows that the Li atom at the 16h site is found to relax to the additional 4c site. The observation confirms the presence of additional sites for Li. For further calculations in the current study, the structure containing the additional sites (4c) has been used. Moreover, the DFT relaxation results show that the Li from densely populated c channels (with number of Li atoms ≥5 within the primitive cell) tend to move into the empty 4c sites upon relaxation, indicating an active role of additional 4c sites to facilitate Li migration. In the densely populated c channel, as shown in Figure 1(a), there exists high electrostatic repulsion

from the averaging process. The approximate average energy barrier can then be written as the following. Ef

Eb approx. =

( )

∑i ∈ N E bi exp − kTi 0

Ef j

( )

∑j ∈ N exp − kT 0

(3)

3. RESULTS AND DISCUSSION 3.1. Structural Details. 3.1.1. Additional Sites. The original crystal structure as proposed by Kamaya et al. had three kinds of sites for Li (16h, 4d, 8f).2 However, classical molecular dynamics study by Adams et al. revealed the presence of an additional fourth kind of site (4c) for Li.5 Later Kuhn et al. also found the presence of additional sites for Li experimentally.6 In the current study, as we are investigating the Li-diffusion mechanism, it is of utmost importance that the structure is well resolved and all the relevant sites for Li 29004

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The Journal of Physical Chemistry C between the Li atoms, and hence, the extra Li at the 16h site is pushed out of the c channel to the neighboring vacant 4c site. The Li at the additional 4c sites is found to relax into sparsely populated c channels (with number of Li atoms ≤2) as shown in Figure 1(b). Such a motion involving the 4c sites gives rise to the possibility of correlated hop between c channel and ab plane providing cross-channel ion migration pathways. Such kind of correlated hop is discussed in detail in section 3.3.3. 3.1.2. Liquidity of the Li Sublattice. On relaxing the proposed structure in DFT as discussed in the previous section, we observe another interesting fact other than the presence of the additional Li sites. We find that Li atoms, on relaxation, move from their proposed 16h sites to the positions which are nearly equidistant along the c-directional channels. The origin of these atomic shifts can be explained by the electrostatic repulsion between the closely spaced Li atoms which leads to the relative liquidity of the Li sublattice when compared with other sublattices (i.e., Ge−P sublattice and S sublattice, which hardly moves during structural relaxations). The liquidity of the Li sublattice along c-directional channels is shown in Figure 2(a),(b). Figure 2(a) shows the nonequidistant spacing between the consecutive Li atoms along the c channel in the unrelaxed configuration, while Figure 2(b) shows the spacing to be nearly equidistant in the relaxed structure. Clearly, the relaxation causes the Li atoms to move in order to achieve a quasi-equidistant configuration indicating the liquidity of the Li sublattice along the c-directional channels. 3.2. Factors Affecting Elemental Li Hops. The diffusion in LGPS was found to be highly anisotropic by Kamaya et al.2 DFT studies by Du et al. also showed a c directional energy barrier of 0.23 eV along the c channel and 0.37 eV along the ab plane.8 Due to the reported anisotropy in diffusivity, first we perform a rigorous study to understand the underlying factors and mechanisms behind the observed diffusion process along the c channel for all possible surrounding environments for the hopping Li. Moreover, we calculate the weighted average of energy barriers (weighted according to the formation energy of the initial configuration of the hop as per eq 2) for the entire set of possible configurations to correlate them to the effective energy barrier observed for the Li-diffusion process. In the following subsections, we first discuss the factors that affect a Li hop along the c channels, namely, the Li population in the c channel (section 3.2.1) and the local environment (i.e., the Li-vacancy ordering in the adjacent sites of the c directional channels) (section 3.2.2). These two factors create different mechanisms for Li hop along the c-directional channels as well as in the ab plane, which we discuss in section 3.3. The corresponding diffusion energy barriers, their dependence on local environment, and their effective contribution to the overall diffusion process are discussed in section 3.4. 3.2.1. c Channel Population. The c channel population is constituted by 16h and 8f sites (see Figure 3). There are four c channels in the primitive unit cell. There are four 16h sites per c channel. According to experimentally observed fractional occupancy of 0.466 for 16h sites,6 two out of four 16h sites can be occupied in a c channel. Further we observed that the two Li atoms cannot be nearest neighbors as they tend to repel each other and attain a quasi-equidistant configuration upon structural and electronic relaxation. There are two 8f sites per c channel. The experimentally observed fractional occupancy of 0.745 for 8f sites6 leads to 1.5 Li atoms in 8f sites per c channel. To decorate the structure, it implies that half of the c channels contain 1 Li atom in 8f sites, and the other half contain 2 Li

Figure 3. Channel population and environment. Different Li sites are marked in different colors: 16h in green, 8f in violet, 4c in blue, and 4d in red. The fraction of color shows the fractional occupancy of the respective site. 16h and 8f sites are along the c channel. Two out of four 16h sites are occupied per c channel, while either one or both 8f sites are occupied along the c channel. The total c channel population can be either four as in the left c channel or three as in the right c channel. The environment around the c channel is determined by occupancy of 4c (blue) and 4d (red) sites (numbered (1−8)). Three out of the four 4c sites are occupied, giving (4C3) 4 possibilities. Either three or four out of four 4d sites are occupied giving 5 (4C3 + 4C4) possibilities. Hence, the total number of environments owing to the occupancy of 4c and 4d sites is 20 {4C3 × (4C3 + 4C4)}.

atoms in 8f sites. Hence, combining the population in 16h and 8f sites, the total c channel population is 3 in half of the c channels and 4 in the other half (as shown in Figure 3). As we will see later in section 3.4, the energy barriers for diffusion in these differently populated c channels are different. Previous studies, as in the case of one by Du et al.,8 have not considered the effect of c channel population, especially the transport in c channels with higher Li density. 3.2.2. Local Environment around a Hopping Li. The environment around an atomic hop along the c-directional channel gets altered with the altered occupancies of 4c and 4d sites around it (numbered 1−8 in Figure 3). While considering the Li atoms along one of the c-directional channels, the Li atoms in neighboring c-directional channels are twice as far as compared to Li atoms in 4c and 4d sites in between the two consecutive c-directional channels. Hence, it is expected that the occupancies of other c channels will be of negligible influence on the energetics associated with the aforementioned atomic hop. Therefore, in determining the environment dependence of energy barrier, occupancies of only 4c and 4d sites are considered in the current study. The fractional occupancy of Li in 4c sites is measured to be 0.817.6 Therefore, the number of Li in 4c sites can be estimated to be 3 out of 4 within the limited precision due to only 4 available sites in the primitive cell. Three Li atoms and one vacancy in 4c sites lead to 4 (4C3) possible configurations. Similarly, the occupancy of Li in 4d sites is experimentally measured to be 0.866.6 Therefore, the number of Li in 4d sites can be estimated to be either 3 or 4 out of 4, which gives rise to 5 (4C3 + 4C4) possible configurations. So, the total number of different environments owing to the occupancy of neighboring 4c and 4d sites is 20 (4C3 × (4C3 + 4C4) = 20). 3.3. Atomic Hop Mechanisms for Li Diffusion. 3.3.1. c Channel Hops. Previous studies show that Li atoms along the c 29005

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The Journal of Physical Chemistry C channel migrate in a synchronized way.4,8 Moreover, we observe that the isolated motion of a single atom along the c channel is not feasible because it encounters huge repulsion from its neighboring atom along the c channel and bounces back to the initial configuration prior to the hop. In the case of synchronized motion of Li atoms along the c directional channel, there are two possibilities: when the primitive cell c channel contains 3 Li atoms and when it contains 4 Li atoms. When the channel has 3 Li atoms, each of the atoms moves in a synchronized way to their next vacant site along the c channel as shown in Figure 4. When the channel has 4 Li all the Li

neighboring 16h site is occupied, the incoming Li will repel the existing Li. In this case, diffusion can occur if the two Li atoms move in a synchronized way: 4c → 16h, 16h → 4c, which is shown in Figure 6. The mechanism was named as knock-off

Figure 6. γ hop. The simultaneous movement of Li atoms (black) from 4c → 16h and 16h → 4c site.

mechanism in previous studies,8 meaning one Li atom in the 4c site knocks off Li in the 16h site to another 4c site. In the current study we call it γ hop. In the case the neighboring 16h site is unoccupied, another hop arises which we discuss separately in the next section (section 3.3.3). 3.3.3. c-ab correlated Hop. In the current study, we find another important hop mechanism, (we name it δ hop), which consists of a correlated motion of Li atoms between the c channel and ab plane as shown in Figure 7. Via a correlated

Figure 4. α-hop. The Li atoms along the c channel move in a synchronized way to their respective next vacancies.

moves cooperatively to the position of the next atom along the c channel, as shown in Figure 5. We name these hops as α hop and β hop, respectively. 3.3.2. ab Plane Hop. The Li in the 4c site is responsible for the cross-channel diffusion. The Li in the 4c site can move to the 16h site in the neighboring c channel. The 16h site in the neighboring c channel can be occupied or unoccupied. If the

Figure 5. β-hop. The initial and final configurations for β-hop are shown in (a) and (b), respectively. The four Li atoms (1,2,3,4) (shown in black) move cooperatively and displace the next atom along the c channel (4 → 3 → 2 → 1 → 4).

Figure 7. δ hop shows correlated motion of atoms along the c channel and in the ab plane. 29006

DOI: 10.1021/acs.jpcc.6b10967 J. Phys. Chem. C 2016, 120, 29002−29010

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The Journal of Physical Chemistry C hop, a c channel, having 4 Li atoms within the primitive cell dimension, can lose one Li atom to a 4c site. This correlated hop results in the motion of Li atoms along the c-directional channel as well as in the ab plane. Furthermore, δ hop along with a consecutive γ hop allows exchange of lithium atoms across the c-directional channels causing alteration of Li density in the c channels. In section 3.3.1, we have discussed that the Li-transport mechanism changes significantly with the change in Li density along a c-directional channel. Here we observe that a particular c channel can act as a fast diffusing path for lithium atoms compared to a neighboring c channel, while a δ hop followed by a γ hop can completely reverse the order. It is to be noted that both γ and δ hops signify the role of 4c sites in the cross-channel diffusion. Without the inclusion of 4c sites, the diffusion studies are bound to miss the significant contribution of the ab plane in the diffusion process. Singular diffusion in one directional channel is less likely to sustain repeated charging and discharging cycles that an electrolyte of a Li-ion cell goes through. Due to the simple antisite point defects, the most mobile c channels can get blocked, inhibiting the diffusion process significantly. In such a situation, the crosschannel diffusion keeps the diffusion process alive, and therefore, inclusion of 4c sites in the LGPS structure is so crucial in understanding the fast diffusion mechanism. 3.4. Diffusion Energy Barriers. The most probable initial configuration for a hop is the ground state. Therefore, the hops initiated from the ground state are the major contributors in the overall ion transport process. In Figure 8, we are showing the

difference in directional diffusivities, and the observation was attributed to the lower difference of activation energies.3,5 The method relied on calculation of the activation barrier solely based on the directional diffusivity. We find this method insufficient to elucidate the underlying atomic hop mechanisms as it overlooks (i) the distribution of migration energies with formation energies of initial configurations for the hops and (ii) the possibility of a correlated motion which contributes to the diffusivities along different directions simultaneously. In our study, we find a third possibility of correlated hop which results in both c-directional and ab plane motion of the Li ions, which is associated with the least energy barrier of all, the δ hop (∼170 meV) (see Figure 8). As the initial configuration is the ground state and the energy barrier is the least corresponding to the δ hop, the probability of this hop is the highest among all at 0 K. Therefore, we expect the δ hop to be the most significant contributor in the ion transport process at very low temperatures. Earlier experimental10 and computational studies3,5 based on MD simulations overlooked this particular mode of transport due to the methodological limitations. With the information available from the discovery of the δ hop in the current study, the targeted NMR studies can confirm the activation energy barrier for δ hop experimentally and verify our prediction. As diffusion is reported to be predominant along the cdirectional channels, we perform a rigorous calculation of diffusion energy barrier for α and β hops for all possible environments resulting from the occupancy of neighboring 4c and 4d sites. Figure 9(a) shows the distribution of the migration energy for different α hops along with the relative formation energy of the corresponding initial configuration leading to that hop. The 0 K relative formation energy for any configuration is calculated relative to the ground state configuration having the same composition. A hop is more probable at some elevated temperature when the migration energy is less. Moreover, the less the relative formation energy of the initial configuration prior to the hop, the more the chance of occurrence for that hop. Therefore, the probability of any hop is conditional to the probability of the initial configuration. In Figure 9(a), the bottom-left corner belongs to the most probable hop and the top-right corner to the least probable hop. A similar argument applies to the β hops, which are represented through Figure 9(b). We note that for both α and β hops, both of which are along the c-directional channels, the variation of relative formation energy for the initial configuration is between 0 and 550 meV. The migration energy barrier for the α hops varies from 60 to 340 meV, while the same corresponding to the β hops varies between 100 and 350 meV. The overall effect of this variation on the ion transport process at a particular temperature can be computed by finding the ensemble average of the migration energies as defined by eq 2. The calculated average represents the effective energy barrier corresponding to the class of hop (α or β). These effective barriers or ensemble averages are shown by the green diamonds in Figure 9(a)−(d). The lowest two configurations of Figure 9(a) and Figure 9(b) have the maximum contribution to the ensemble average. These two configurations are significantly more stable (by about 150 meV) than the next higher energy configurations in relative energy scale. Hence the subset of these two environments has been considered to calculate the energy barrier for γ and δ hop as shown in Figure 9(c) and Figure 9(d). The overall average

Figure 8. Energy variation for Li-ion migration along the migration pathway for various kinds of hops corresponding to the ground-state configuration of LGPS.

energy variation along the migration coordinate for all kinds of hops (α, β, γ, and δ) originating from the ground-state configuration, as calculated by the NEB method. Clearly, the c channel hops (α and β) are associated with significantly lower energy barrier (∼180 meV) compared to the pure ab plane hop, the γ hop (∼370 meV). The finding corroborates with the earlier findings of lower activation energy for c-directional hops.3,5,8 Nevertheless, the low degree of anisotropy of the diffusivity, as observed by the experiments, is ill-explained because of the large difference in activation energies. The earlier studies based on the molecular dynamics simulations found less 29007

DOI: 10.1021/acs.jpcc.6b10967 J. Phys. Chem. C 2016, 120, 29002−29010

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Figure 9. Effect of local environment on energy barrier of different types of Li hop: (a) α, (b) β, (c) γ, and (d) δ. The different local environments owing to the occupancy of 4c and 4d sites shown in Figure 3 and as shown inset (e) are represented as a square with 8 triangles. The color coding of the triangle represents: blue: 4c occupancy, red: a 4d occupancy, and white: a vacancy. The y axis represents the relative energy of the initial configuration corresponding to a particular local environment relative to the ground state, and the x-axis shows its associated energy barrier. The labels for the y axes for the plots (c) and (d) are the same as in (a) and (b). The green diamonds represent the coordinate for the ensemble average formation energy and the ensemble average energy barrier for the entire set of configurations considered for the particular type of hop.

anisotropy is significantly reduced due to the correlated δ hops. The global average of the activation energies, as calculated by eq 2, relates directly to the observed macroscopic diffusivity of the material and is shown by the pink diamond in Figure 10. The value (239 meV) as computed by the current study closely matches with the experimentally observed value by Kamaya et al. (240 meV).2 The remarkable quantitative match verifies the underlying mechanisms as discussed here and settles the debate about the degree of anisotropy in LGPS.

energy barrier or ensemble average has been calculated by eq 3 considering this subset only of the most stable configurations. The ground-state configuration is the most probable one at 0 K and hence the corresponding hops. At elevated temperature, the higher energy configurations become accessible, and the hops corresponding to these higher energy initial configurations contribute to the overall transport process. Therefore, an ensemble average provides the effective activation barrier for a class of hop. In Figure 10, we show these average values corresponding to room temperature for all the hops (α, β, γ, and δ). Here, we see that the c-directional hops (α and β) are the lowest (211 and 198 meV), and the hop corresponding to the motion in the ab plane only (γ) is the highest (337 meV) in activation energy. The correlated δ hop (210 meV) is close to the c-directional hops in activation energy. The physical implication of these activation energies would be an anisotropic diffusion with c-directional dominance, while the degree of

4. CONCLUSIONS The migration energy barrier of Li atoms along the c channel depends on the c channel Li population, local environment of Li surrounding the c channel, and the type of hop. With the variation in the occupancies of the 4c and 4d sites surrounding the c channel, the migration energy barrier along the c channel varies from 50 to 350 meV. The broad spectrum of energy 29008

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Figure 10. Ensemble average for different types of hop: α hop (green triangle), β hop (blue left triangle), γ hop (orange right triangle), and δ hop (red gradient). (The green colored diamonds of Figure 9(a)−(d) are shown here together for comparison. The points are not actual relative formation energies or energy barriers of any configuration but are statistically averaged quantities of the data represented in Figure 9(a)−(d), calculated by eq 2 and eq 3, and are plotted on the same set of axes. The data on the y axis are the average formation energy of the set calculated by the formula similar to eq 2 and eq 3.) The overall average is shown by the pink diamond.

cross-channel diffusion as well as the importance of correlated δ hop as discovered by the current study allow us to generate intuition about the diffusion process and its sustenance with cycling for the whole class of LGPS-type solid electrolytes.

barrier suggests that there exist local environments which correspond to significantly low energy barrier (close to 100 meV). Although the configurations with low energy barrier are less stable, they may be accessible at higher temperatures. The overall average energy barrier calculated in the current study is 239 meV which is in remarkable agreement with the experimentally observed value by Kamaya et al.2 Among the c channel hops, the β hop has slightly less average energy barrier than the α hop, showing that the synchronized motion along a 4 Li-populated c channel is easier than that along a 3 Lipopulated c channel. The average energy barrier for a correlated hop (δ hop) is significantly lower than a pure ab plane hop and is comparable to the c channel hops (α and β hop) which indicates that the correlated mechanism of motion between the c channel and ab plane is easier than pure ab plane motion and is as dominant as the pure c channel motion. The newly found correlated hop mechanism between c channel and ab plane can work for Li diffusion even when a part of the c channel or ab plane gets blocked, thus providing alternate pathways for diffusion. Further, the existence of a correlated mechanism in LGPS is due to the presence of a large percentage (37.5%) of vacancies (12 vacancies out of 32 sites in the primitive cell). Vacancies, especially in directions perpendicular to the c channels, allow for correlated motion between the c channel and the ab plane. The occurrence of a large percentage of vacancies in Li 5 La 3 Ta 2 O 12 (76.19%) and Li 7 La 3 Zr 2 O 12 (53.33%) suggests the possibility of a correlated hopping mechanism in these structures as well. Nevertheless, the large percentage of vacancy serves only as a precondition to the correlated hops, and we have not studied the energetics associated with such hops in the aforementioned materials. The prediction of a new hop and its related energy barrier can be experimentally verified by the targeted NMR studies. The present investigation includes information on both the energy barrier and the probability of the atomic arrangement conducive to the hop which enables us to calculate the range of diffusivities instead of a single value in each relevant direction. Henceforth, the confusion about the degree of anisotropy is settled. Moreover, the role of additional sites in

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Corresponding Author

*E-mail: [email protected]. ORCID

Arihant Bhandari: 0000-0002-2914-9402 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Arihant Bhandari gratefully acknowledges the financial support from MHRD (Ministry of Human Resource Development), Govt. of India. Jishnu Bhattacharya acknowledges the initiation grant from IITK (Indian Institute of Technology, Kanpur) for carrying out the study. We acknowledge the High Performance Computation facility by IIT Kanpur.

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REFERENCES

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