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Origin of the Phase Transition in Lithium Garnets Fei Chen, Junyang Li, Zhifeng Huang, Ying Yang, Qiang Shen, and Lianmeng Zhang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b10911 • Publication Date (Web): 12 Jan 2018 Downloaded from http://pubs.acs.org on January 12, 2018

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The Journal of Physical Chemistry

Origin of the Phase Transition in Lithium Garnets

Fei Chena,*, Junyang Lia, Zhifeng Huanga, Ying Yangb, Qiang Shena, Lianmeng Zhanga a

State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology, Wuhan, 430070, China

b

Department of Engineering Structure and Mechanics, Wuhan University of Technology, Wuhan, 430070, China * Corresponding author, e-mail address: [email protected]

Abstract In this study, molecular dynamic and density functional theory based simulations were performed to obtain a better understanding of the origin of phase transition in garnet-type Li7La3Zr2O12 solid electrolyte. The phase transition coincided with the lithium redistribution among all sites. With the investigation of lithium distribution and dynamics, we found one temperature dependent lithium migration pathway in lithium garnets. Lithium ions exhibited uniformly 3-dimensional diffusion in cubic LLZO, while the lithium diffusion in tetragonal LLZO was mainly in a and b direction. The constrained diffusion in c direction in tetragonal LLZO could be ascribed to the blocking effect of 16f sites which was found to be thermodynamically more stable than tetragonal 8a and 32g sites through density functional theory based calculations. Besides, the stabilizing effect of supervalent doping on cubic phase was also studied through Ta-doped LLZO. Further site occupancy investigations indicated

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that supervalent doping introduced lithium vacancies and reduced octahedral sites (tetragonal 16f and 32g sites) occupancy due to site energy preference. The reduced octahedral sites occupancy weakened the blocking effect of tetragonal 16f sites, promoted the lithium redistribution, and eventually lowered the phase transition temperature.

1 Introduction With the introducing of solid electrolyte, all-solid-state batteries obtain great improvements in safety and energy density compared to flammable liquid organic electrolyte based lithium ion battery. Acting as a combination of electrolyte and separator, solid electrolyte offers mechanical strength to prevent lithium dendrite growth, high lithium ionic conductivity, chemical and electrochemical stability in contact with lithium anode and most cathode materials.1–3 Among reported candidates, lithium garnets exhibit capable performance with ionic conductivity vary from 10-6 to 10-3 Scm-1 and excellent stability with metallic lithium.4,5 One of the lithium garnet families being widely studied is Li7La3Zr2O12 (LLZO) with cubic structure (space group 3).5 Dodecahedral LaO8 and octahedral ZrO6 form the 3D framework by edge-sharing. Lithium ions are randomly and partially distributed in tetrahedral 24d sites and octahedral 96h sites as shown in Figure 1(c). One LiO4 tetrahedron connects with four LiO6 octahedron. One LiO6 octahedron is surrounded by two LiO4 tetrahedron. LiO4 tetrahedron and LiO6 octahedron are connected with each other by face-sharing and finally form a 3D continuous lithium


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transport pathway as shown in Figure 1(e). However, pure LLZO generally crystallize to thermodynamically more stable tetragonal phase (space group 4 / ) at room temperature.6 The transition from cubic to tetragonal phase is called tetragonal distortion, as shown in Figure 1(f), which transforms the cubic 24d sites into fully occupied 8a sites and unoccupied 16e sites. Cubic 96h sites are also transformed into two fully occupied 16f and 32g sites.7,8 This tetragonal distortion has little influence on the framework, but greatly affects lithium distribution and lowers the ionic conductivity by 2 orders of magnitude.5,6 Several studies from both experimental and theoretical investigations had been focusing on the stabilization of the cubic phase. The most efficient way to obtain room temperature stable cubic phase is introducing lithium vacancy into lithium sublattice by supervalent doping, i.e. Al3+ (Li site)9–11, Ga3+ (Li site)12–14 and Ta5+ (Zr site)15,16, as well as summarized in recent review17. This method has been widely used in the preparation of cubic lithium garnets. Jalem et al.18 performed molecular dynamic simulations to study the cubic phase stabilizing effects of gallium. Their results showed that gallium doping can reduce the phase transition temperature of garnets. Cubic phase could be stable in room temperature when phase transition temperature was lower than room temperature. Similar result was also obtained by Bernstein et al.7 in the determination of effect of composition on phase transition of LLZO. The decrease of phase transition temperature can be regarded as a sign of the stabilization of cubic phase. Thus, knowing the mechanism of the phase transition from tetragonal to cubic



phase in lithium garnets is necessary for the better understanding of the stabilization of cubic phase and the further design of lithium garnets with higher ionic conductivity. Previous reports19–24 showed that lithium distribution changed from order to disorder during the phase transition from tetragonal to cubic phase in lithium garnets. Adams’s group captured the whole transition process of LLZO from both neutron diffraction and molecular dynamic simulations with their own Morse-type force-field21,25. Their results showed an approximately agreement and revealed that tetrahedral sites occupancies increased with increasing temperature, while octahedral sites occupancies decreased in the meantime. Bernstein et al.7 performed first-principle-based molecular dynamic simulations to investigate phase transition process of LLZO. Only the simply shift from 8a to 16e sites was observed during the phase transition. Wang and his co-workers24 performed neutron diffraction and molecular dynamic calculations to study the local lithium distribution in the Li7-xLa3Zr2-xTaxO12 (x = 0-2) series. Their results showed an increasing tendency of lithium ordering as lithium content increased. The local lithium distribution suggested that tetrahedral sites lithium occupancy was generally lower than that obtained from neutron diffraction and the exclusion principle. The X-ray and neutron diffraction performed by Logéat et al.23 also showed that lithium ions were promoted from tetrahedral sites to octahedral sites during Ta-doping which might reduce electrostatic repulsion. In the view of thermodynamics, the phase transition in LLZO is an entropy-driven process19,20, during which lithium ions were distributed among all sites.


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Supervalent doping reduced lithium content and disorder lithium sublattice12,21, which promoted the shift of lithium distribution from order to disorder and eventually reduced the free energy7. Efforts have been made to study the relationship between lithium distribution and ionic conductivity. Burbano et al.26 ascribed the low ionic conductivity of tetragonal LLZO to concerted ionic motion around a closed-loop due to the strong lithium ordering by performing molecular dynamic simulations. Such closed cyclic diffusion didn’t make any contribution to ionic conductivity. By applying first-principles-based calculations, Meier et al.27 revealed two different Li-ion migration mechanism. In tetragonal LLZO, high ordering of Li-ion results in a fully collective motion, which requires higher activation energy. In cubic LLZO, single-ion jump mechanism brings much lower energetic cost and higher ionic conductivity compared with the fully collective mechanism in tetragonal LLZO. Similar concerted migration mechanisms were also obtained by Jalem et al.28 in their FPMD simulations. Chen et al.29 performed molecular dynamic simulation to elucidate the lithium diffusion mechanism in LLZO. Their results showed that lithium exhibits uncorrelated Poisson-like diffusion in the cubic LLZO and correlated diffusion in the tetragonal LLZO. They further confirmed that the uncorrelated diffusion is resulted from the weak site dependence of lithium diffusion in the cubic LLZO. Morgan30 studied the correlation effects for lithium garnets by lattice-gas Monte Carlo simulations and found that the correlation in lithium garnets is reproduced from both site energy differences and nearest-neighbor repulsion. Furthermore, the ionic conductivity can be



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maximized by tuning the mobile-ion stoichiometry. Above studies26–30 indicated that different lithium distributions could greatly influence the ionic conductivity by exhibiting different lithium migration mechanism. In general, the studies of the phase transition of LLZO were concentrated in the lithium redistribution during the phase transition from tetragonal to cubic phase and the effects of lithium distribution on ionic conductivity. But the investigations about the energetic origin of the lithium redistribution and how lithium distribution affect the migration mechanism were still empty. In this paper, we performed molecular dynamic simulations and density functional theory based calculations to reveal the origin of the phase transition in Li7La3Zr2-xTaxO12 (x=0-1, LLZ-Tax). The relation between the lithium distribution and the lithium migration mechanism during the phase transition of LLZO were first studied. Then, the lithium sites energy differences were calculated to understand the energetic origin of the lithium redistribution during the phase transition. Besides, the effects of supervalent dopants on lithium distribution and phase transition temperature in doped-LLZO were also characterized.

2 Methods 2.1 Interatomic potentials Empirical interatomic potentials consisting of long-range coulombic interaction and short-range Buckingham interaction, as given by eq. 1, were used to describe atomic interactions. Φ = −

−

+


!"#

(1)

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where , $ , % are empirical force-field parameters for interacting & and ' ions, () is the free space permittivity, is the distances between & and ' ions, and * represents the charges of & ion. All the force-field parameters except for the Buckingham-type potential parameters of Ta-O pair were taken from the literature31. Interactions parameters between Ta and O were fitted against the experimental structure of Ta-doped LLZO24 by the GULP package32. The force-field parameters used in this work were listed in Table 1. The cutoff distances were set to 12 Å for all short-range Buckingham interaction. The Ewald summation were used to summarize long-range coulombic interaction. Table 1. Force-field parameters. Buckingham potential parameters Species

Charge (e)

Li-O

(eV)

$ (Å)

% (eVÅ6)

1.00

1087.29

0.260

0.00

La-O

2.50

2075.26

0.326

3.25

Zr-O

2.65

1650.32

0.311

5.10

Ta-O

3.65

713.831

0.359

0.36

O-O

-1.65

4870.00

0.267

77.00

2.2 Lithium arrangements To balance the accuracy and computational efficiency, a 2×2×2 supercell containing 576 available Li sites were used in this study31. In order to obtain the natural distribution of lithium ions, the disordered lithium distribution of the initial



model was obtained from the ordered lithium distribution from thermal relaxation. Tetragonal and cubic LLZO shared the same framework structure, but their crystal structure was a little bit different. For the convenience of the simulation of the whole phase transition process, tetragonal LLZO (Figure 1(a)) was transformed into one cell (Figure 1(b)) whose framework had the same fractional coordinates with that of the cubic phase (Figure 1(c)). After this symmetric operation, Ta ions uniformly and symmetrically distributed in Zr site and replaced one neighbor octahedral site Li ion (32g or 16f) with vacancy simultaneously to maintain electrical neutrality.

2.3 Molecular dynamic simulations Molecular dynamic simulations were performed from 300 K to 1400 K with 2 fs time step and 1 atm pressure using the GULP package32. In the first step, initial models were heated under constant number, pressure and temperature (NPT) ensemble for 50 ps to determine cell sizes at target condition. In the second step, using cell sizes obtained from NPT simulations, we performed constant number, volume and temperature (NVT) ensemble simulations for 1000 ps to obtain dynamic features. Thermostat and barostat were used with a relaxation time of 0.05 ps and 0.25 ps, respectively. Trajectories were sampled with a time step of 0.1 ps for NPT run and 1 ps for NVT run. Ionic conduction and local structure were analyzed from atomic trajectories with number density, pair distribution function (PDF33) and mean squared displacement (MSD). Number density was calculated by dividing the cell into a 300×300×300 mesh


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grid. Then, the number of Li ions in each grid were counted and averaged over all time steps. To ensure that every Li was counted in the evaluation of the occupancies of lithium in different symmetrical position, the distances between Li and every stable Li sites in cubic LLZO (24d and 96h) were calculated in fractional coordinate. Li would be labeled as its nearest stable Li site. Occupancies were the number of Li averaged over all time steps and the number of stable sites. MSD was calculated over all simulation times from following definition 〈,-(/)12 〉 =

4

∑4 7-6 (/ + /) )12 − ,7-6 (/) )12 〉 8〈,

(2)

where 9 is the number of atoms. The self-diffusivity was defined as : = lim>→@ A

2B>

〈,-(/)12 〉C

(3)

where is the dimension of diffusion. According to above equation, when it reaches diffusive regime, the slope of log〈,-(/)12 〉 versus log / should be 1. For these simulations in this study who reaches diffusive regime, : is calculated by the linear fitting to the dependence of 〈,-(/)12 〉 over 2/.

2.4 Density functional theory based calculations First-principle calculations were performed in 1×1×1 cell using the Perdew-Burke-Ernzerhof generalized gradient approximation (GGA-PBE)34 and the Projector Augmented Wave (PAW)35 method via the Vienna ab initio simulation package (VASP)36. The kinetic energy cutoff of 520 eV and 4×4×4 k-points mesh was used for all the calculations. All structures were relaxed in two steps. In the first step, the cell parameters were relaxed with fixed fractional coordinates. In the second step,



the atoms were allowed to relax with fixed cell parameters. The energy of the relaxed structures were then calculated.

3 Result and discussion 3.1 Phase transition and lithium conduction modelling To better understand the phase transition from tetragonal to cubic phase and the stabilization of cubic phase, molecular dynamics simulations of LLZO with Ta-dopants concentration varying from 0 to 2 were performed with empirical force-field parameters fitted against experimental structures as mentioned in section 2.1. The temperature dependence of lattice parameters of Ta-doped LLZO captured from NPT simulations are shown in Figure 2(a). The phase transition process of LLZO can be captured from the changes of lattice parameter. When phase transition from tetragonal to cubic phase occurs, the lattice parameters transform from tetragonal feature (a = b ≠ c) to cubic feature (a = b =c). The rapidly merging of lattice parameters between 850 and 950 K can be regarded as a sign of the phase transition from tetragonal to cubic phase. Results show that the phase transition temperature decreases with the increasing Ta-dopants concentration. When Ta-dopants concentration is larger than 0.25, no tetragonal phase is observed. The phase transition temperature is so sensitive to lithium concentration that the stabilization process of the cubic phase is observed in a small concentration windows. It should be noted that, due to the inherent limitation of the Buckingham-type interatomic potential, it’s hard to get the right phase transition temperature and also the critical Ta-dopants


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concentration for the stabilizing of cubic phase. Figure 2(b) shows the linear dependence of lattice parameters on dopants concentration, known as the Vegard’s law. Our simulations match well with the experimental lattice parameters15,24,37–40. The occurrence of phase transition can also be seen from diffusion properties of lithium ions. Figure 2(c) shows the MSD of lithium ions from NVT simulations range from 300 K to 1400 K with an interval of 100 K in LLZO. According to eq. 3, the slope of MSD is 6: for 3-dimensional diffusion. The larger the slope, the greater the self-diffusivity. The negligible MSD of tetragonal phase represents its low diffusivity compared to that of cubic phase. The MSD of 900K locates at the middle part of MSD of tetragonal and cubic phase. The data point that is clearly different from the point around and changes rapidly help us locate the phase transition process. Figure 2(d) shows the dependent of log〈,-(/)12 〉 on log /. As mentioned in methods section, when it reaches diffusive regime, the slope of log〈,-(/)12 〉 versus log / should be 1. The simulation time used in this paper is long enough to reach diffusive regime for high temperature simulations. At lower temperature, diffusive regime can be reached in the last hundreds of picoseconds of the whole simulation time. For the simulations at 300 to 500 K, MSD keep constant values throughout the simulations, suggesting the oscillation behavior of lithium ions at this temperature range. Consequently, the MSD data of 300 to 500 K are not used to calculate the self-diffusivity. In general, the empirical force-field parameters used in this work are able to capture the phase transition process and obtain lithium ion diffusion properties. With Ta doping, the stabilization process of cubic phase is also observed. In the following



sections, we will focus on the relationship between lithium distribution, diffusion pathway and the phase transition in LLZO firstly. Then, the effect of Ta doping on the stabilization of cubic phase are investigated on the basis of the understanding of the phase transition.

3.2 Lithium distribution in LLZO It can be confirmed that the phase transition of LLZO is accompanied by the lithium redistribution among all tetrahedral and octahedral sites. In order to understand lithium distribution in LLZO, Li-Li pair distribution function are calculated for all the simulations as shown in Figure 3(a). Similar approaches have been made by Adams et al.21 with their own Morse-type force-field and Wang and Klenk19,20 with a Buckingham-type interatomic potential. Li-Li pairs are separated with each other in a distance larger than 2 Å, which is the same as Adams et al.21 obtained. This distribution feature results from the electrostatic repulsion suggests that two neighboring sites in a distance smaller than 2 Å, i.e. two distorted octahedral sites in one octahedron, tetrahedral sites and its nearest distorted octahedral sites, cannot be occupied simultaneously. Physically reasonable lithium arrangements, as described by O’Callaghan et al.41, are shown in Figure 3(b): I, two occupied tetrahedral sites and unoccupied octahedral site, II, one occupied tetrahedral site and an occupied distorted octahedral site away from the tetrahedral site, III, two unoccupied tetrahedral sites and an occupied octahedral site. Combining the above three basic arrangements, all possible local lithium arrangements that contribute to Li-Li pair distribution function


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are shown in Figure 3(c) and Figure 3(d). These local arrangements contain all the cases where lithium distance is less than 5 Å. It should be noted that, in cubic structure at 1400 K (Figure 3(c)), arrangement I offers Li-Li distance of 4.049 Å and Li-Li PDF shows that there is a valley. As a result, the first two combinations, I+I and I+II, should not exist or be relatively few, Li-Li PDF is mainly contributed from the last four combinations: (II+II)1, (II+II)2, II+III and III+III. After that, from a combination standpoint, there is no difference in local lithium arrangement between cubic and tetragonal phase: two structures are both come from the combination of arrangement II and III. This result is corresponding to statistical result for octahedral centered clusters performed by Klenk et al.20. To figure out the difference of Li-Li PDF between tetragonal and cubic phase, the distances between lithium ions in all local arrangement are measured and marked on Figure 3. In the distance less than 5 Å, cubic phase has two Li-Li PDF peaks while tetragonal has three peaks. The first peak of cubic PDF is contributed by a large amount of Li-Li pairs, the distances of these pairs are 2.389, 2.539, 2.704, 2.938, 3.396, 3.092, 3.356 Å. During the tetragonal distortion of cubic phase, the first five distances change to 2.532, 2.683 and 2.645 Å, which contributes to the first peak of tetragonal PDF. 3.092 Å increases to 3.696 Å and contributes to the second peak of tetragonal PDF. Due to the splitting of angle between two arrangements II in combination (II+II)1, 3.566 Å splits into 3.794 Å and 4.473 Å, which contributes to the second and third peak of the tetragonal peak. Unlikely, Adams et al.21 observed two peaks (2.35 and 2.8 Å) in the Li-Li PDF in tetragonal phase, between which there is a valley. However, in our



measurements, no Li-Li pair with a distance of 2.35 and 2.8 Å is observed. In general, our simulations match well with experimental structure refined from neutron diffraction24 and suggest that (i) the splitting and widening of PDF peaks result from the distortion of lithium ring caused by tetragonal distortion, (ii) the first peak of Li-Li PDF moves to larger distance during temperature decrease from 1400 to 300 K. This result indicates that tetragonal structure can offer a distribution that allows lithium ions separate from each other in a larger distance, which may reduce Li-Li repulsion and make tetragonal structure more thermodynamically stable. To intuitively observe the lithium distribution in the crystal structure, lithium density maps in different temperatures are shown in Figure 4. While the differences of density map between adjacent temperatures are relatively small, only four temperature points with typical feature are shown, which are 500 K, 700 K, 900 K and 1100 K. Lithium ion located at different symmetrical sites are marked with different colors. The unoccupied 16e site is surrounded by two 16f and two 32g sites and marked as pink hollow circle in Figure 4(a) and (e). At 500 K (Figure 4(a) and (e)), lithium ions prefer to locate at their stable sites and only oscillations are observed. It’s worth noting that, in this temperature, 32g sites lithium ions may jump to 16e sites, stay for a little while and then jump back. When temperature increase to 700 K, the overlapping of some 8a (red part) and 32g sites (green part) are observed. The overlapping sites form a continuous pathway throughout the simulation box as marked by black solid line in Figure 4(b) and (f). Such pathway is considered to be lithium migration pathway. Furthermore, these migration pathways exhibit spiral structure


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whose axis appears to be only in a and b direction. As temperature increases to 900 K (Figure 4(c) and (g)), at which the phase transition begins, all stable sites including 16f sites are connected and preliminarily form a 3-dimensional lithium migration network. After the phase transition has occurred, lithium ions distribute more evenly over each site as mapped in Figure 4(d) and (h). From above discussion, we can conclude that (i) the phase transition has little influence on lithium local arrangements, but affects the Li-Li pair distance a lot, (ii) phase transition from tetragonal to cubic is accompanied with the lithium redistribution among all lithium sites rather than from tetragonal 8a sites to tetragonal 16e sites7,20. Moreover, there may exist different lithium migration pathway before and after the phase transition. In the next section, the evolution of lithium migration pathway with phase transition from tetragonal to cubic will be analyzed and mapped by combining site occupancy and local structure.

3.3 Evolution of lithium migration pathway in LLZO Figure 5(a) shows the temperature dependence of occupancy of different sites. The local structure of lithium ions in tetragonal phase are shown in Figure 5(b). Combining occupancy and local structure, the evolution of lithium migration pathway can be divided into four stages by temperature. In the first stage, from 300 to 500 K, 8a and 16f sites remain fully occupied, indicating their oscillation feature in this stage. This can also be visualized from the density map in Figure 4(a) and (e). As mentioned in the last section, lithium ions shift



from 32g sites to 16e sites. This shift forms a local migration between 32g-16e-32g as mapped in green in Figure 5(b) but doesn’t contribute to ionic conductivity. In the second stage, there is a shift of lithium ions from 8a and 32g to 16e. The vacancies in 8a, 32g and 16e sites are responsible for the forming of lithium migration pathway. The local structure of lithium network (Figure 5(b)) shows that these three sites can be connected into a continuous pathway, 8a-32g-16e-32g-8a (the yellow part). Ab-initio molecular dynamics simulations by Meier et al.27 shows a similar migration pathway in tetragonal LLZO with which we described in Figure 6(c). However, the migration pathway they proposed (16f-32g-8a-32g-16f) cannot form a continuous pathway, which may be because of the neglect of 16e sites. The spiral structure, or helix, of above pathway are shown in Figure 6(d). Figure 6(b-c) shows the side view of two pathways in Figure 6(a). In the direction of the axis of the helix, 8a, 32g and 16e sites are arranged in circles as shown in Figure 6(a) and marked with blue solid line. One helix can be considered as 1-dimensional lithium migration pathway and contributes to the ionic conduction in the axis direction. The axis of the helix is parallel to a or b direction and is perpendicular to c direction. The helix-pathways in a and b directions are perpendicular to each other and intersect in 8a sites. The lithium ions in two helix-pathways can be exchanged in 8a sites, which is the only contribution to ionic conduction in c direction. To compare the differences of diffusivity of lithium ions between c direction and ab plane, the diffusivities of them were calculated from the slope of MSD (As shown in Figure S1) and plotted as Figure 5c. As shown, in this stage, ionic conduction is mainly contributed by the


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conduction in ab plane. The ionic conduction in c direction is relatively small but nonnegligible, and is blocked by 16f sites. Thus, the lithium conduction in tetragonal can be considered as 2-dimensional-like conduction. In the third stage, when 16f sites finally participate in lithium conduction, a 3-dimensional lithium migration network is formed since there are no blocking ions. Density map (Figure 4(c) and (g)) confirm the formation of a 3-dimensional lithium migration network. Occupancies show that phase transition and lithium redistribution from 16f sites to other sites begin simultaneously. Meanwhile, the gap of diffusivity between ab plane and c direction is getting smaller as shown in Figure 5(c), suggesting the blocking effect of 16f sites may be the key factor that prevents the phase transition from tetragonal to cubic phase. In the final stage, the lithium redistribution among all lithium sites is completed. The occupancy of tetragonal 8a and 16e sites, 16f and 32g sites become the same in value. The first two sites merge into cubic 24d sites and the last two sites merge into cubic 96h sites. This is corresponding to the experimental refined structure. Figure 5(c) shows that lithium conduction in ab plane and c direction has no difference in this stage. To obtain the thermodynamic origin of the differences of the temperature that lithium participates in conduction between different sites, DFT simulations of lithium defective LLZO model is performed. The lithium vacancy is obtained by substitute Zr4+ with Ta5+. Four lithium defective models, which are denoted as V8a, V16f, V32g-near and V32g-far (Figure 7(a)), are built in the consideration of the relative position of



lithium ion and Ta5+. The energies of the defective models are shown in Figure 7(b). The higher the energy of the defected structure, the harder it is to form. Since 16f vacancy has higher energy, lithium vacancy should prefer to locate at other sites, like 32g. Thus, the order of stability is 16f > 8a > 32g, which explains the temperature differences that lithium participates in conduction between different sites. Since tetragonal phase and cubic phase differ only in the lithium distribution, the phase transition is actually one process of lithium redistribution from tetragonal distribution to cubic distribution. The evolution of lithium distribution and migration pathway is derived from the site stability. With the fact that 32g and 8a sites lithium does not affect the linear dependence of lattice parameter on temperature, while 16f sites lithium promotes the merging of lattice parameters, we believe that 16f sites affect the phase transition from tetragonal to cubic phase by blocking the lithium redistribution in c direction.

3.4 Effect of Ta-doping on lithium distribution From above discussions, we known that the phase transition in garnets is actually the redistribution of lithium atoms. In this section, the effect of supervalent doping on lithium distribution is investigated. Figure 8 shows the evolution of occupancies of LLZ-Tax with different Ta content as a function of temperature. According to the separation of different diffusion stage as what has been done in section 3.3, the occupancies are colored differently. Similar with the temperature dependence of lattice parameter, Ta doping shifts the whole occupancy evolution to low temperature


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and keeps the order of participation of different sites in lithium conduction (32g > 8a > 16f), which confirms the four diffusion stages mechanism discussed above. Besides, the phase transition temperature is sensitive to lithium content and decreases rapidly with the decreasing lithium content. It also suggests that cubic phase can be obtained when Ta-dopants concentration fall into a small range from 0.125 to 0.25, which corresponds to experimental fact that a small amount of dopants can stabilize the cubic phase14,23,40. Figure 9(a) shows the change of occupancy during the stabilization of cubic phase. Ta doping can lower the occupancy of 16f sites and 32g sites simultaneously. The merging of occupancies is also observed. After the merging of occupancies, the redistribution among tetrahedral and octahedral sites are still going on. Figure 9(b) shows the occupancies of tetrahedral and octahedral sites as a function of Ta dopants concentration. Red and green area represent the permitted occupancy range21 of tetrahedral and octahedral sites derived from electrostatic repulsion41, respectively. Both neutron diffraction5,6,24,42–46 and our molecular dynamic work shows a strong trend that octahedral site occupancy exhibits the minimum value of permitted occupancy. It should be noted that the permitted minimum occupancies of octahedral sites are calculated with the assumption that lithium ions prefer to locate at tetrahedral sites. The increasing tetrahedral sites occupancies can also push lithium ions from undistorted octahedral site to distorted octahedral sites due to electrostatic repulsion. Thus, when the occupancy of tetrahedral sites is larger than 0.5, all lithium ions only locate at the tetrahedral sites and the distorted octahedral sites (see Fig. 3(b)41). The



assumption is corresponding to our DFT results that 8a sites (tetrahedral sites) are more stable than 32g sites (distorted octahedral sites). Hence, 16f sites (undistorted octahedral sites) is responsible for the formation of tetragonal phase by blocking lithium diffusion in c direction. Figure 10 shows the temperature dependence of lithium ion diffusivity in Ta-doped LLZO. Ta doping could improve lithium diffusivity by lowering phase transition temperature and keeping the linear trend of diffusivity in Arrhenius plot to room temperature. Further increasing the Ta-dopants concentration will decrease the diffusivity. The ionic conductivity is the result of the combination of lithium concentration, lithium diffusivity and the correlation effect. Our results can also be used to explain the stabilization of cubic phase by low-valent doping like Y3+ 47,48. The increased lithium concentration derived from low-valent doping forces lithium ions to fill in the unoccupied 16e sites. This increases the occupancies of tetrahedral sites and push lithium ions from octahedral sites to distorted octahedral site, which weakens the blocking effect of octahedral sites.

4 Conclusion In summary, combined molecular dynamics and density function theory based calculations, we investigated the origin of phase transition from tetragonal to cubic phase and the stabilization of cubic phase in lithium garnets. First, temperature dependence of lattice constants and mean squared displacement were employed to


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confirm that our simulations were able to capture the phase transition and model lithium conduction. Second, lithium distributions were explored via Li-Li pair distribution function and lithium density map. Li-Li pair distribution function showed that the phase transition didn’t change the lithium local arrangements. Moreover, tetragonal phase could provide a distribution scheme with enlarged Li-Li distance. Lithium density map described to us a 2-dimensional-like lithium migration pathway in tetragonal phase. Third, combining with lithium local structure, the structure and the formation of the different migration pathway were carefully analyzed from the temperature dependence of sites occupancies. We divided the evolution of lithium migration pathway into four stages: oscillation, local migration, 2-dimensional-like migration and 3-dimensional migration. DFT results showed that the evolution was the result of sites energy difference. The blocking effect of 16f sites lithium ions was the main factor that hampered the transition from tetragonal to cubic phase. Finally, the effects of supervalent doping on lithium distribution were investigated. The phase transition temperature was sensitive to lithium content and decreased rapidly with the decreasing lithium content. Ta-doping also promoted the shift of lithium from octahedral sites to tetrahedral sites, which turned out to be the result of site energy preference. This work deepens the understanding about the phase transition and the stabilization of cubic phase in lithium garnets.

Acknowledgements This work is financially supported by the National Natural Science Foundation of



China (No. 51472188, and 51521001), Natural Research Funds of Hubei Province (No. 2016CFB583), Fundamental Research Funds for the Central Universities in China, State Key Laboratory of Advanced Electromagnetic Engineering and Technology （ Huazhong University of Science and Technology), National Key Research and Development Program of China (NO.2017YFB0310400) and the “111” project (No. B13035).

Supporting Information Figure S1 shows the mean square displacement of lithium ions in LLZO along a, b and c direction, respectively.

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Figure captions Figure 1. Crystal structure of (a) tetragonal LLZO, (b) transformed tetragonal LLZO and cubic LLZO. Lithium arrangement in (d) tetragonal and (e) cubic LLZO. (f) The loop structure of lithium arrangement in both tetragonal and cubic LLZO. Figure 2. (a) Temperature dependence of lattice parameter of Li7-xLa3Zr2-xTaxO12 (x = 0 - 2). (b) The Vegard’s law. Experimental lattice parameter are taken from references (Wang24,40, Thompson38, Buschmann39, Hamao37 and Li15). (c) Mean squared displacement of lithium ion at temperatures from 300 to 1400 K with an interval of 100 K. (d) Log-log plot of mean squared displacement. Figure 3. (a) Li-Li pair distribution function as a function of temperature. (b) Physically reasonable lithium arrangements.41 Local lithium arrangements in (c) cubic and (d) tetragonal LLZO derived from 1400 K and 300 K NPT run, respectively. Figure 4. Lithium density map derived from (a), (e) 500 K, (b), (f) 700 K, (c), (g) 900 K and (d), (h) 1100 K NVT run. Lithium ion at 8a, 16f and 32g sites are colored red, blue and green, respectively. The observed migration pathway in tetragonal phase are marked in solid black line in (b) and (f). Figure 5. (a) Temperature dependence of site occupancies in LLZO. (b) Local structure of lithium network. (c) Arrhenius plot of the total diffusivity and the diffusivity in ab plane and c direction. Figure 6. Lithium arrangement in half cell and the schematic diagram of migration pathway in a (solid black line) and b direction (solid blue line) from (a) (010) and (b-c)



(001) crystal face in tetragonal LLZO. (b) and (c) differ in the choosing of half cell. (d) The schematic diagram of the 8a-32g-16e-32g-8a helix migration pathway in tetragonal LLZO. Figure 7. (a) The schematic diagram and (b) the relative energy of four lithium defective model. Figure 8. The temperature dependence of lithium site occupancy in Li7-xLa3Zr2-xTaxO12. (a) x = 0.03125, (b) x = 0.0625, (c) x = 0.125, (d) x = 0.25, (e) x = 0.375, (f) x = 0.5. Figure 9. The lithium site occupancies at (a) 400 K and (b) 1400 K as a function of Ta dopants concentration. Red and Green area represents the permitted occupancy range of tetrahedral and octahedral sites derived from electrostatic repulsion, respectively. Open symbols are taken from neutron diffraction data in references5,6,24,42–46. Filled symbols represents the average occupancies of cubic tetrahedral and octahedral sites. Figure 10. Arrhenius plot of the self-diffusivity of Ta-doped LLZO as a function of Ta dopants concentration.


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Figure 1



Figure 2


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Figure 3



Figure 4


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Figure 5



Figure 6


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Figure 7



Figure 8


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Figure 9



Figure 10


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TOC Graphic



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Origin of the Phase Transition in Lithium Garnets - The Journal of

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