Solid-State Electrolytes - American Chemical Society

Article pubs.acs.org/JPCC

Solid-State Electrolytes: Revealing the Mechanisms of Li-Ion Conduction in Tetragonal and Cubic LLZO by First-Principles Calculations Katharina Meier,* Teodoro Laino, and Alessandro Curioni IBM Research Zürich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland ABSTRACT: In this study, we applied first-principles-based calculations, such as ab-initio molecular dynamics simulation, metadynamics, and nudged-elastic band calculations, to successfully identify the mechanisms responsible for the considerable difference in ionic conductivity between the tetragonal and the cubic phases of LLZO (Li7La3Zr2O12)a promising candidate for use as a highly Li-ion conductive solid-state electrolyte in Li-based batteries. Whereas in tetragonal LLZO the motion of Li ions is of fully collective nature or synchronous, we identified an asynchronous mechanism dominated by single-ion jumps and induced collective motion in cubic LLZO. The latter mechanism is possible at considerably lower energetic cost. The calculated energetic barriers that represent the two distinct mechanisms show good agreement with experimental values. Moreover, we were able to map the different mechanisms to the structural features of the particular polymorphs.

■

INTRODUCTION With the steady decrease of natural resources, the search for renewable energy sources has become indispensable and concomitantly triggered the need for improved energy storage technologies. When it comes to electric vehicles, lithium−air batteries are considered the most advanced technology for electrical energy storage.1−4 Lithium−air batteries could potentially provide an up to 10 times greater specific energy than state-of-the-art Li-ion batteries used in most electronic devices today.5−8 Over the past years, a great deal of research has been devoted to development of efficient, rechargeable Li/ O2 cells.7−10 A key component in designing highly efficient Li/ O2 batteries is the Li-ion-conducting electrolyte that enables the transport of Li ions between the negative Li-metal anode and the positive porous carbon cathode that promotes diffusion of oxygen from the environment into the electrochemical cell. In the past, electrolytes based on liquid organic solvents such as ethylene carbonate or propylene carbonate were most commonly used in aprotic lithium−O2 batteries.11 However, recent experimental3,12−14 and theoretical15,16 evidence showed that organic electrolytes degrade upon discharge, resulting in an efficiency loss of the electrochemical cell. Electrolyte degradation is mainly caused by the strong oxidative character of the discharge product, Li2O2. Due to the generally high flammability of organic solvents, use of organic liquid electrolytes bears a considerable safety risk. Additional drawbacks in the deployment of liquid electrolytes are, for example, limitations for battery miniaturization and the risk of environmental contamination upon improper disposal. The importance of finding suitable electrolyte candidates for Li/O2 batteries has considerably stimulated research efforts.17−19 Recently, attention has been directed toward solid-state electrolytes that combine desired properties such as high safety (nonflammable), ease of device fabrication, and low cost.8,19−24 © 2014 American Chemical Society

The main challenge in development of solid-state Li/O2 batteries is to find novel materials for solid-state electrolytes that combine the characteristics of safety and thermodynamical, chemical, and electrochemical stability with high Li-ion conductivity. Roughly 10 years ago, Thangadurai et al.25−27 identified garnet-like lithium oxides as a new class of crystalline Li-ion conductors with potential use as highly Li-ionconductive solid-state electrolytes. Recently, a new member of this garnet family was synthesized.28 The zirconiumcontaining lithium−lanthanum−oxide (LLZO) with structural formula Li7La3Zr2O12 is probably the most promising candidate for use in solid-state batteries to date thanks to its low cost28,29 and thermal stability. Most interestingly, it exhibits extraordinary chemical and electrochemical stability while possessing high Li-ion conductivity.28,29 In contrast to conventional garnets that typically crystallize in cubic form, the thermodynamically stable phase of LLZO is tetragonal at room temperature.30,31 The tetragonal phase exhibits a relatively low Li-ion conductivity (σtetra ≈ 10−6 S/ cm).30 Only at high temperature a cubic symmetry with an up to 2 orders of magnitude higher Li-ion conductivity (σcubic = 10−4 S/cm) is observed.28,29,32,33 Depending on the purity of the sample, the temperature for reversible phase transition lies approximately between 400 and 600 K.31,32 Understanding the mechanism responsible for the considerable difference in conductivity between the tetragonal and the cubic phase of LLZO is of great relevance to design highly Li-ion-conductive solid-state electrolytes. Over the past years, several theoretical studies have been published that focus on calculation of bulk ionic conductivities Received: January 9, 2014 Revised: February 18, 2014 Published: March 19, 2014 6668

dx.doi.org/10.1021/jp5002463 | J. Phys. Chem. C 2014, 118, 6668−6679

The Journal of Physical Chemistry C

Article

The tetragonal and cubic polymorphs of LLZO mainly differ in the arrangement of Li ions, i.e., in the Li-ion sublattice.29,30,32 On the basis of single-crystal X-ray diffraction experiments, Awaka and co-workers29 proposed a loop structure as basic descriptor of the Li-ion network. Loop junctions formed by tetrahedral Li sites would connect separate loops to create the 3D pathway proposed for ion conduction.29 For tetragonal LLZO, three different Li sites have been identified:30 a tetrahedral “Li1” (8a) site and two different distorted octahedral “Li2” (16f) and “Li3” (32g) sites. In contrast, in cubic LLZO only two different types of Li sites can be differentiated: a tetrahedral “Li1” (24d) site and a distorted octahedral “Li2” site,29 which can be interpreted as the result of a splitting of the octahedral 48g into two energetically equivalent displaced octahedral 96h sites. While the crystallographic Li sites in tetragonal LLZO are fully occupied,29,30 Awaka et al.29 found a considerably reduced site occupancy (g = 0.35) of the Li2 site and an occupancy of almost one (g = 0.94) for the Li1 site in cubic LLZO. Lower Li-site occupancies in cubic LLZO have also been observed in neutron diffraction studies but with a different ratio of Li1/Li2 occupancy (g(Li1) = 0.56, g(Li2) = 0.44).39 Occupational disorder in the Li sublattice is discussed as a main requirement for high Li-ion conductivity.29,32,34,38,40 However, because of the low scattering intensities of 7Li in X-ray and neutron diffraction experiments an accurate interpretation of the Li-ion population and Li-ion migration pathways by experimental means remains difficult.

and investigation of the Li-ion-conducting pathway in the cubic polymorph of LLZO. Adams and Rao34 performed classical molecular dynamics (MD) simulations applying a Morse-type interaction potential energy function. From simulation trajectories at a wide range of temperatures on a hundreds of picoseconds to nanosecond time scale they derived diffusion coefficients, ionic conductivities, and activation barriers for tetragonal and cubic LLZO in reasonably good agreement with experimental values. A similar approach based on ab-initio MD simulations was presented by Jalem and co-workers.35 However, it might be argued whether 30 ps of simulation time are sufficient to derive meaningful diffusion coefficients. All simulations were carried out at elevated temperature and focused only on the cubic modification of LLZO. Roomtemperature conductivity was obtained via extrapolation. The authors chose a region of the simulation cell involving five Li ions with specifically high ion-jump frequency from the last 5 ps of their simulation at 1273 K to investigate the detailed mechanism of Li-ion migration in cubic LLZO.35 Because of the high temperature, it is not confirmed whether this explicit motion can be associated with low energy and whether it therefore can be considered the most likely mechanism, although all results presented seem to agree well with experiment. Both studies34,35 estimate the energy barrier for ion conduction from Arrhenius plots derived from ab-initio MD simulation trajectories at different temperatures. Xu et al.36 used a static quantum chemical approach. They investigated the minimum energy pathway for Li-ion migration in cubic LLZO using the nudged-elastic band (NEB) method. Finally, Bernstein et al.37 presented a theoretical study that, although not focusing on Li-ion conduction, contributes to the understanding of the structural changes during phase transition from tetragonal to cubic LLZO. Thus far, no study has been presented that investigates and compares the mechanisms of Li-ion migration in both the tetragonal and the cubic modification of LLZO. However, such a comparison could yield crucial insights that help understand the origin of the considerable difference in ionic conductivity between the two phases. Here, we present a systematic theoretical study that investigates the different mechanisms of Li-ion migration in tetragonal and cubic LLZO. High-temperature ab-initio MD simulations have been used to reveal the three-dimensional network of Li-ion-conducting pathways. Metadynamics simulations were applied to obtain a qualitative picture of Li-ion migration at room temperature as well as realistic starting configurations for subsequent quantitative investigations of energetic barriers by nudged-elastic band calculations.

■

COMPUTATIONAL METHODOLOGY All calculations reported in this study have been performed using the CP2K41 code. Density functional theory (DFT) using the Perdew−Burke−Enzerhof (PBE)42 exchange-correlation functional was used throughout. Nuclear cores were represented by Godecker−Tetter−Hutter43 norm-conserving pseudo potentials. For expansion of the Kohn−Sham orbitals, a double-ζ Gaussian basis set with polarization function was used. The auxiliary plane wave basis set was truncated using a density cutoff of 400 Ry. System Preparation. Initial configurations for tetragonal and cubic LLZO were created based on the crystallographic information files icsd_246816.cif and icsd_182312.cif, respectively. Both model structures contained 192 ions and satisfy the stoichiometric formula Li56La24Zr16O96. For the tetragonal model structure with cell dimensions of a = 13.129 Å, b = 13.129 Å, and c = 12.672 Å, all available Li1 (8a), Li2 (16f), and Li3 (32g) sites were filled with Li ions. The resulting arrangement of different Li sites in a loop as proposed by Awaka et al.29 is shown in Figure 1a. In cubic LLZO, the number of available Li sites is considerably larger than the number of Li ions. To account for the high degree of disorder in cubic LLZO, 120 different configurations with respect to the randomly occupied Li positions have been created. Taking the site occupancies published by Awaka et al.29 as reference, 23 Li ions were randomly distributed over 24 tetrahedral sites (corresponds to an occupancy g = 0.96) and 33 Li ions were randomly distributed over 96 octahedral sites (corresponds to occupancy g = 0.34) with the constraint that Li-ion pairs cannot occupy adjacent 96h sites in the cubic cell with dimensions a = b = c = 12.983 Å because of unfavorable electrostatic repulsion. The arrangement of the different Li1 (24d) and Li2 (96h) sites in a loop structure as proposed by Awaka et al.29 is shown in Figure 1b.

■

REVIEW OF STRUCTURAL FEATURES OF LLZO LLZO is a representative of garnet-like lithium oxides. Conventional garnets have the composition A3B2C3O12, where the cations A, B, and C occupy crystallographic sites with 8-fold, 4-fold, and 6-fold oxygen coordination, respectively.38 In LLZO, lanthanum occupies the cubic A sites whereas zirconium is contained in the octahedral C sites. Garnets typically exhibit fully occupied tetrahedral B sites. However, they are capable of hosting an excess number of cations38 as in the case of LLZO. LLZO possesses a surfeit of lithium cations that reside in some of the usual tetrahedral garnet sites as well as in octahedral oxide environments that are unoccupied in conventional garnets. 6669



Article

Figure 1. Loop arrangement of different Li sites: tetrahedral Li1 site (yellow), octahedral Li2 (pink), and Li3 (green) sites: (a) tetragonal LLZO, (b) cubic LLZO.

After initial structure preparation, each geometry was relaxed using the Broyden−Fletcher−Goldfarb−Shanno optimization algorithm as implemented in CP2K.41 The resulting Li-ion network of tetragonal LLZO is shown in Figure 2. For cubic LLZO, the structure with the lowest potential energy was chosen from the set of 120 optimized geometries for subsequent calculations. The Li-ion network representing the configuration with lowest energy is given in Figure 3. Lattice parameter optimizations for the geometry-relaxed tetragonal and cubic model structures were performed using a conjugate gradient algorithm and an external pressure of 1 bar. A slight cell expansion with lattice parameters of a = 13.207 Å, b = 13.209 Å, and c = 12.659 Å for tetragonal and a = b = c = 13.065 Å (with cell symmetry fixed) for cubic LLZO was observed. However, the deviations from the corresponding experimental lattice parameters are less than 1% and can thus be considered in very good agreement with experiment. Therefore, all calculations reported below were performed using the experimental lattice parameters. Occupation analysis was performed using a lithium−oxygen cutoff distance of 2.1 Å for tetrahedral and 3.1 Å for octahedral sites, taking into account periodic boundary conditions. Investigating Conduction Pathways. To obtain a realistic picture of possible conduction pathways in tetragonal and cubic LLZO, we performed MD-based simulations and NEB calculations as will be discussed in the following.

Figure 3. Arrangement of Li ions in cubic LLZO model structure. Tetrahedral Li1 and distorted octahedral Li2 sites are shown in yellow and pink, respectively. Solid: Occupied Li sites in generated model structure with minumum energy. Transparent: Available Li sites according to cubic space group (Ia3̅d).

High-Temperature Ab-Initio Molecular Dynamics. Simulations were conducted in the canonical ensemble (NVT). The temperature was kept constant at 1500 K using the Nosé− Hoover thermostat44−46 with a chain length of three and a time constant of 0.1 ps. The volume was set according to the cell size of the tetragonal and cubic simulations cells, respectively (see above). Newton’s equations of motion were integrated using the second-order Velocity-Verlet algorithm47 and a time step of 1 fs. Energy gradients were derived from DFT calculations within a Born−Oppenheimer MD scheme. Trajectories were saved every 10 fs. Metadynamics. Metadynamics48 was employed only to obtain qualitative estimates of possible low-energy migration pathways at room temperature. To enhance the sampling of Liion configurations, the root-mean-square deviation (RMSD) with respect to the Li-ion positions was chosen as collective variable. Simulations were performed at 300 K. The Gaussian height was 0.05 eV.

Figure 2. Arrangement of Li ions in tetragonal LLZO model structure. Tetrahedral Li1 and distorted octahedral Li2 and Li3 sites are shown in yellow, pink, and green, respectively: (a) view along z axis; (b) view along x axis. 6670



Article

Minimum Energy Pathway Calculation. Li-ion conduction pathways were optimized using the climbing image NEB method49,50 as implemented in CP2K.41 To probe the conduction pathway of interest, 16 images were applied. Geometry-optimized representative configurations from the metadynamics trajectories were chosen as start and end points of the pathway, as will be discussed in detail below. A DIISbased optimization algorithm was applied. Convergence criteria of 0.001 au on forces and 0.0005 au on geometry displacements were used.

■

RESULTS AND DISCUSSION In the following, we will provide a thorough investigation of possible Li-ion migration pathways in tetragonal and cubic LLZO. Li-Ion Conduction in Tetragonal LLZO. High-Temperature MD: Exploring the 3D Li-Ion Network. High-temperature ab-initio MD was used to investigate possible Li-ion migration pathways. Fixed cell size and limited simulation time prevents the occurrence of a phase transition to the cubic polymorph. Figure 4 shows the 3D network of pathways

Figure 5. Radial distribution function (g(r)) for Li−Li pairs averaged over 20 ps of ab-initio MD simulation started from tetragonal LLZO model structure. Periodicity of the simulated unit cell is taken into account.

Here, we applied metadynamics48 to facilitate transitions between different low-energy configurations of Li ions and thus enable sampling of Li-ion migration at room temperature. Figure 6 shows a representation of Li-ion positions that have

Figure 4. Li-ion positions visited during 20 ps of ab-initio MD simulation at 1500 K started from the tetragonal LLZO model structure.

explored by Li ions in tetragonal LLZO during 20 ps of MD simulation at 1500 K. The observed network of crossing pathways reflects the 3D loop structure suggested by Awaka et al.29 Loop junctions correspond to tetrahedral Li1 sites (Figure 1a). The average distance for neighboring Li-ion pairs along the pathway is around 2.5 Å (first peak in Figure 5). Reported experimental values are 2.5 Å for Li1−Li3 distances and 2.6 and 2.7 Å for Li2−Li3 distances.30 A high-temperature MD simulation samples an ensemble of configurations that might include configurations of the Li-ion arrangement that are energetically unfavorable at room temperature. However, it gives a valuable qualitative picture of the three-dimensional motion of Li ions in the simulated unit cell. Metadynamics: Qualitative Estimate of Possible LowEnergy Ion-Conducting Pathways. In the course of a 10 ps unbiased MD simulation at 300 K, Li-ion migration could not be observed. This is mainly because of the energetic barriers between different low-energy configurations of the system. The energetic barrier for Li-ion migration that was experimentally determined to be 0.5 eV30 is too large to be overcome during only 10 ps of simulation at 300 K. Sampling of those rare events requires a considerably longer simulation time or use of specific sampling-enhancement techniques.

Figure 6. Li-ion positions of tetragonal LLZO as observed from metadynamics simulation at 300 K. Transparent gray: all Li-ion positions visited during 10 ps run. Solid: starting configuration of Li ions. Blue: ions involved in collective motion. Arrows indicate the direction of motion.

been visited during 10 ps of the simulation run. We observed a continuous pathway explored by the Li ions numbered 1−6. Two different Li-ion arrangements dominated in residence time. This observation is reflected in the times series of the collective variable given in Figure 7a (left). One preferred arrangement is characterized by a RMSD around 25 au, whereas the other arrangement is characterized by a RMSD of around 100 au. As can been seen from the right panel of Figure 7a, these arrangements also represent minima in the corresponding free energy profile. Representative snapshots of the two Li-ion arrangements extracted from the simulation trajectory at 2.5 and 5.6 ps reveal the type of Li sites associated with the observed pathway (Figure 7b). The composition of stable Li sites (Li2−Li3−Li1−Li3−Li2) along the pathway corresponds to part of the loop depicted in Figure 1a. While 6671



Article

Figure 7. (a) Time series of collective variable from metadynamics simulation at 300 K starting from tetragonal LLZO (left panel) and corresponding free energy profile (right panel). (b) Representative Li-ion configurations of the observed pathway extracted from metadynamics trajectory at 2.5 (lower presentation) and 5.6 ps (upper presentation). Pink and green indicate octahedral Li2 and Li3 sites, respectively; yellow indicates tetrahedral Li1 site.

migration from one to the other preferred Li-ion arrangement occurs, each Li ion moves to the position of its preceding neighbor. The individual RMSDs of each Li ion participating in the observed movement are plotted in Figure 8. At

Energetic Barriers for Li-Ion Conduction. On the basis of our qualitative findings from metadynamics simulation, we calculated the minimum energy pathway for Li-ion migration between the two different Li-ion arrangements depicted in Figure 7b. The positional change corresponds to a collective motion of all 6 Li ions involved. Figure 9 shows the relative

Figure 8. Root-mean-square deviation (RMSD) for individual Li ions of tetragonal LLZO evaluated for 10 ps of metadynamics simulation at 300 K: (a−f) Li-ion indices 1−6 as marked in Figures 6 and 7a.

Figure 9. Potential energy profile for minimum energy Li-ion migration pathway in tetragonal LLZO. ΔE: Energy differences (in eV) between individual NEB replicas along the migration pathway and configuration with lowest energy (frame 2).

approximately 4.5 ps of the simulation, all 6 Li ions concomitantly undergo a positional change with average RMSDs between 2.5 and 2.9 Å. These averages agree with the distances between the different Li1, Li2, and Li3 sites reported by Awaka et al.30 The concomitant movement of Li ions clearly shows the collective nature of the Li-ion migration in tetragonal LLZO. As mentioned, metadynamics simulation was used only to obtain a qualitative estimate of possible low-energy migration pathways. Although the free energy profile in Figure 7a (left) indicates an energetic barrier for movement between the two dominant arrangements that already agrees well with the experimental value of 0.56 eV, we do not draw quantitative conclusions at this stage because of the known difficulties in the free energy convergence when applying metadynamics simulation. In our study, metadynamics simulation provided a solid ground for choosing representative configurations for a subsequent quantitative investigation of the energetic barrier for bulk Li-ion migration in tetragonal LLZO.

energies calculated along the optimized pathway. The calculated barriers for Li-ion migration amount to 0.44 (forward movement) and 0.41 eV (backward movement). The experimental barrier is, with a value of ∼0.5 eV,30,31 larger than our calculated lower bound of ∼0.4 eV. However, such a discrepancy is still within the error margin that can be expected from a DFT calculation applying the NEB method. Furthermore, our reported barriers refer to ionic motion in the bulk and do not account for grain-boundary contributions. Given these aspects, the calculated energetic barrier of 0.44 eV can be considered a reasonably good estimate for Li-ion migration in tetragonal LLZO. Our results clearly show that the relatively high energetic barrier for movement of Li ions in tetragonal LLZO is due to a mechanism of synchronous collective ionic motion. This can be interpreted as a direct consequence of the complete ordering of the (almost equidistant) Li-ion sublattice with full occupation of all available Li sites. Li-Ion Conduction in Cubic LLZO. Structural Characterization of the Starting Configuration. In the cubic unit cell of 6672



Article

LLZO, 56 Li ions can in principle occupy 24 tetrahedral (24d) and 96 distorted octahedral sites (96h). However, when a Liion pair occupies neighboring 96h sites, the close distance of 1.6 Å29 is likely to cause unfavorable Coulomb repulsion between the two ions. As a consequence, the potentially available number of octahedral sites reduces to 48. With 56 ions distributed over 72 available Li sites, the number of possible Liion configurations is on the order of 1015. Sampling all possible Li-ion configurations for choosing a representative starting configuration for subsequent investigations is by far too demanding computationally. However, it is crucial to take into account the high disorder and large number of vacancies when one wants to set up a realistic computational model to study cubic LLZO. In our approach, we introduced some degree of sampling by generating 120 structures in which 56 Li ions are placed randomly in 72 available sites. Figure 10 Figure 11. (Top) Relative energies (eV) of 120 energy-minimized cubic LLZO model structures with randomized Li positions. (Bottom) Occupation numbers (= occupied sites divided by available sites) of tetrahedral (blue) and octahedral (red) sites for each of the energyminimized model structures. Considering that two neighboring octahedral 96h sites cannot be occupied concomitantly, a value of 48 available octahedral sites has been used.

High-Temperature Ab-Initio MD. The 3D network of Li-ion conduction pathways in cubic LLZO obtained from 10 ps of high-temperature ab-initio MD simulation is given in Figure 12a. During 10 ps of simulation we observed a much wider range of visited Li-ion positions along the pathway, leading to a more continuous representation of the network than observed for tetragonal LLZO during even double the simulation time. This could be an indication of the considerable higher ionic mobility in cubic LLZO. The absence of short Li−Li distances (Figure 12b) confirms that no occupancy of neighboring 96h sites occurs. In fact, the distribution of Li−Li distances with a peak around 2.5 Å is similar to that of tetragonal LLZO (Figure 5). Our observation does not support the hypothesis that shorter Li−Li distances in cubic LLZO could be responsible for the higher ionic conductivity.32 Metadynamics: Qualitative Estimate of Possible LowEnergy Ion-Conducting Pathways. Figure 13 visualizes the Liion-conduction pathway for cubic LLZO as obtained from 10 ps of metadynamics simulation at 300 K. We identified six Li ions that mainly contributed to the observed continuous pathway. The time series of the collective variable (RMSD of Li ions) and the corresponding free energy surface are shown in Figure 14. The collective variable strongly fluctuates, and the free energy exhibits a minimum that largely differs from the experimental values suggested for energetic ion-conduction barriers. However, the results obtained do not reflect a converged free energy. Achieving full convergence of the free energy in metadynamics simulation can be very cumbersome and requires a considerable amount of simulation time. Moreover, the algorithm relies strongly on a proper choice of the collective variable that should uniquely define the reaction coordinate of the process under investigation. The cumulative RMSD of all Li ions chosen as collective variable might also involve jump events that do not directly contribute to the ionic conduction. Similarly to the tetragonal system, here we also use only metadynamics to provide realistic estimates of Li-ion configurations that characterize a possible conduction pathway at room temperature and can be used as an initial guess for

Figure 10. Population analysis for 120 energy-minimized cubic LLZO model structures with randomized Li positions.

presents a population analysis of the potential energies of the set of optimized model structures. Assuming a Boltzmanndistributed ensemble at room temperature, the structures with lowest energy (left tail of the distribution in Figure 10) will dominate. Zero-point vibrational energy corrections did not affect the overall picture of energetically ordered structures substantially. For subsequent investigations, we chose the structure with the lowest potential energy (corresponding to index 81 in Figure 11). Controversial information has been published regarding the occupancy of tetrahedral and octahedral sites in cubic LLZO. Awaka et al.29 suggest an almost 100% site occupancy of tetrahedral Li1 sites. For the set of 120 starting configurations, we took this suggestion as an initial guess and observed that, in all cases, only about 60% of the 24 Li1 sites remain occupied after geometry optimization (Figure 11, bottom). In turn, occupation of the 48 octahedra per unit cell increases to about 80% (corresponds to about 40% of 96h sites). For example, the structure with lowest potential energy (index 81) has 11 (46%) occupied tetrahedral Li sites and 45 (94%) occupied octahedra (corresponds to 47% occupied 96h sites), see Figures 3 and 11. Our observation agrees well with recent experimental and theoretical findings.35,36,39 Summarizing results from refs 32,35,36, and 39 and our study, it can be concluded that in cubic LLZO occupancy of octahedral Li2 (48g/96h) sites is generally preferred over occupancy of tetrahedral Li1 (24d) sites. 6673



Article

Figure 12. (a) Li-ion positions visited during 10 ps of ab-initio MD simulation at 1500 K starting from a cubic LLZO model structure. (b) Radial distribution function (g(r)) for Li−Li pairs averaged over 10 ps of ab-initio MD simulation started from a cubic LLZO model structure. Periodicity of the simulated unit cell has been taken into account.

Figure 13. Li-ion positions in the simulated unit cell of cubic LLZO visualized for two different angles of view. (Transparent gray) Positions of Li ions (plotted every 0.15 ps) visited during 10 ps of metadynamics simulation at 300 K started from a cubic LLZO model structure. The more frequent a position is visited the darker the color. (Solid) Li-ion positions of a geometry-optimized trajectory snapshot at 8.98 ps. Ions dominanting characterization of the conduction pathway are indexed 1−6 and colored blue.

minimum energy pathway (MEP) was determined using the NEB method. The energy profile of the corresponding MEP is shown in Figure 15a. It exhibits two peaks, suggesting two transition states along the pathway. The first barrier amounts to 0.23 eV, a reasonable value if compared with reported experimental values (0.2651 to 0.34 eV28) and considering the error margin of the NEB method. However, the calculated values for the second barrier 0.49 and 0.76 eV are too large to be representative of the ion conduction in cubic LLZO. It seems that a synchronous collective motion as assumed for the transition from the initial to the final configuration (Figure 15b) does not reflect the ion-conduction mechanism in cubic LLZO. This assumption is supported when comparing Figures 8 and 16. While in Figure 8 all Li ions show fully synchronous jumps, more variety is observed in Figure 16. The time window of almost 5 ps that separates the end-point configurations of the MEP assuming collective motion in cubic LLZO covers several jumps as can be seen in Figure 16. Moreover, configurational changes between the start and the end point also involve positional fluctuations of Li ions not contributing directly to the conduction pathway under investigation. For this reason, we decided to investigate the end point configurations corresponding to a smaller time window of the metadynamics trajectory. Figure 17a shows the energy profile corresponding to the MEP that connects geometry-optimized structures of snapshots taken at 3.6 and 4.3 ps. The pathway exhibits a considerably smaller energetic barrier with values of 0.18 (forward) and 0.13 eV (backward). Investigating the optimized frames along the MEP, we observed a jump of ion 2 from the octahedral position into the neighboring octahedral

Figure 14. Time series of collective variable from metadynamics simulation at 300 K starting from a cubic LLZO model structure (left) and corresponding free energy profile (right).

subsequent calculations of the energetic barrier for Li-ion conduction. Energetic Barriers for Li-Ion Conduction. As first step in investigating energetic barriers for migration of Li ions in cubic LLZO, we assumed a synchronous collective mechanism for Liion migration as it was observed for tetragonal LLZO (see above). We chose two configurations from the metadynamics trajectory that showed a large difference in the cumulative RMSD (collective variable, Figure 14, left), namely, the configurations at 4.02 and 8.98 ps of the simulation trajectory. After geometry optimization of the two initial structures, the 6674



Article

Figure 15. (a) Potential energy profile for the minimum energy Li-ion migration pathway in cubic LLZO assuming fully collective motion of Li ions 1−6. ΔE: Energy differences (in eV) between individual NEB replicas along the migration pathway and configuration with lowest energy (frame 16). (b) Arrangement of Li ions 1−6 and corresponding sites at the start (left) and end (right) point of the NEB calculation: (turquoise) tetrahedral Li1 sites, (gray) octahedral Li2 sites.

frames. In fact, minimum energy pathway calculations strongly depend on both choices. However, because the authors36 did not describe their choice in detail, our conclusions remain speculative. We then separated the ionic motion along the initially assumed pathway (Figure 15) into single-ion movements and calculated the barrier for each of those movements. The individual movements were chosen based on visual inspection of the trajectory configurations obtained from metadynamics. The first movement describes a single ion jump of Li ion 1 out of its octahedral position into the neighboring octahedral vacancy (see structures A and B, Figure 19). Again, we only observe a single energetic barrier (Figure 18) and no noticeable residence in the face-sharing tetrahedral vacancy between the two octahedra. The next step describes the transition of Li-ion 5 from octahedral into its face-sharing tetrahedral neighbor position (B → C, Figure 19), which triggers the movement of ion 6 from one distorted octahedral 96h site to the other energetically equivalent 96h site. The transition B → C has a quite shallow energetic barrier (0.09 eV) and an end point (C) with larger potential energy than the starting point. An alternative jump with slightly larger energetic barrier is described by transition B → Cb (Figure 19, see also Figure 18). Transition C → D corresponds to a movement of ion 4 from its octahedral position to the edge-sharing octahedral neighbor site which is followed by the jump of ion 3 from its tetrahedral site into the freed octahedral vacancy (D → E). This movement triggers a concomitant displacement of ion 1 into a distorted tetrahedral position (E). The final transition (E → F) results in a Li-ion arrangement in the conduction pathway

Figure 16. Root-mean-square deviation (RMSD) for individual Li ions (nos. 1−6) of cubic LLZO evaluated for 15 ps of metadynamics simulation at 300 K.

vacancy without triggering a concomitant movement of ions 3 or 4. Only some cooperative motion of ions 5 and 6 was involved. The ionic Li positions that were identified along the MEP are shown in Figure 17b and 17c. These findings foster the assumption that the ionic motion is not synchronous collective in cubic LLZO. In contrast to the NEB results by Xu et al.,36 we did not observe a two-barrier pathway for movement of a Li ion from one octahedral position to the neighboring octahedral position with “residence” in a Li1 site. This can be rationalized from a different choice of starting configurations and interpolating 6675



Article

Figure 17. (a) Potential energy profile for minimum energy Li-ion migration pathway in cubic LLZO from the Li-ion arrangement of the geometryoptimized snapshot at 3.6 ps to that at 4.3 ps of the metadynamics trajectory at 300 K. ΔE: Energy differences (in eV) between individual NEB replicas along the migration pathway and configuration with lowest energy (frame 0). (b) Arrangement of Li ions at start (left) and end (right) points of NEB calculation. (Transparent) Li positions visited along pathway. View along z axis. (c) View along x axis.

Figure 18. Calculated potential energy (in eV) for each configuration along the Li-ion conduction pathway investigated following a mechanism dominated by single-ion motion. Capital letters indicate the start and end points of each individual NEB calculation. Corresponding Li-ion arrangements along the pathway are depicted in Figure 19. (Red) Alternative ion jump starting from configuration B. 6676



Article

Figure 19. Arrangement of Li ions in the conduction pathway investigated in cubic LLZO. Capital letters denote the start and end point configurations of each NEB calculation (compare Figure 18).

Our findings support the assumption stated above that the Li-ion conduction mechanism in cubic LLZO is not fully collective or synchronous. We observed that single-ion jumps can trigger cooperative or asynchronous motion of other individual ions rather than of a full collection of ions. Calculating energetic barriers for Li-ion conduction based on the assumption of individual ion jumps, we obtained values that are on the order of ∼0.1−0.3 eV and therefore considerably smaller than the barrier calculated for tetragonal LLZO (0.4 eV). This observation indicates a higher ionic conductivity for cubic than for tetragonal LLZO and confirms experimental findings based on single-crystal X-ray diffraction, NMR spectroscopy, and impedance measurements.29,30,32,33

investigated that corresponds to the end-point configuration of our initial NEB calculation presented in Figure 15. It involves a cooperative motion of ion 1 from the distorted tetrahedral to the face-sharing octahedral position and of ion 2 from the distorted octahedral to the face-sharing tetrahedral position. For cubic LLZO, the calculated energetic barriers for each ion displacement (Table 1) are considerably smaller than that obtained for a collective displacement of Li ions (Figure 15). Table 1. Calculated Energetic Barriers for Individual Li-Ion Movements along the Conduction Pathway Investigated in Cubic LLZO transition

ΔEforward (eV)

ΔEbackward (eV)

A→B B→C B → Cb C→D D→E E→F

0.10 0.09 0.24 0.19 0.08 0.22

0.18 0.02 0.19 0.15 0.36 0.13

■

CONCLUSIONS We presented a detailed investigation of the Li-ion conduction mechanisms in tetragonal and cubic LLZO from first principles. High-temperature ab-initio MD simulations validated the 3D network of Li-ion conduction pathways in both tetragonal and cubic LLZO. In cubic LLZO, the conduction pathways were explored considerably faster than in tetragonal LLZO, indicating a higher ionic conductivity. By applying metadynamics to enhance the sampling of Li-ion motion, we were able to reveal an isolated continuous ion conduction pathway for both tetragonal and cubic LLZO at room temperature. On the basis of the pathways observed, we calculated energetic barriers for Li-ion conduction using the nudged-elastic-band method.

In general, we observed that short-distance jumps between face-sharing tetrahedra and octahedra exhibit a slightly smaller energetic barrier than jumps between edge-sharing octahedra (Table 1, Figure 18). Because of the smaller energetic barrier, movements between face-sharing polyhedra might occur concomitantly with a transition between edge-sharing octahedra. 6677



Article

Applications from a Battery Industry Standpoint. Adv. Energy Mater. 2012, 2, 780−800. (5) Tarascon, J. M.; Armand, M. Issues and Challenges Facing Rechargeable Lithium Batteries. Nature 2001, 414, 359−367. (6) Armand, M.; Tarascon, J.-M. Building Better Batteries. Nature 2008, 451, 652−657. (7) Peng, Z.; Freunberger, S. A.; Chen, Y.; Bruce, P. G. A Reversible and Higher-Rate Li-O2 Battery. Science 2012, 337, 563−566. (8) Capsoni, D.; Bini, M.; Ferrari, S.; Quartarone, E.; Mustarelli, P. Recent Advances in the Development of Li-Air Batteries. J. Power Sources 2012, 220, 253−263. (9) Abraham, K. M.; Jiang, Z. A Polymer Electrolyte-Based Rechargeable Lithium/Oxygen Battery. J. Electrochem. Soc. 1996, 143, 1−5. (10) Ogasawara, T.; Holzapfel, M.; Nova, P.; Bruce, P. G. Rechargeable Li2O2 Electrode for Lithium Batteries. J. Am. Chem. Soc. 2006, 128, 1390−1393. (11) Read, J. Characterization of the Lithium/Oxygen Organic Electrolyte Battery. J. Electrochem. Soc. 2002, 149, A1190−A1195. (12) Mizuno, F.; Nakanishi, S.; Kotani, Y.; Yokoishi, S.; Iba, H. Rechargeable Li-Air Batteries with Carbonate-Based Liquid Electrolytes. Electrochemistry 2010, 78, 403−405. (13) Xu, W.; Viswanathan, V. V.; Wang, D.; Towne, S. A.; Xiao, J.; Nie, Z.; Hu, D.; Zhang, J.-G. Investigation on the Charging Process of Li2O2-based Air Electrodes in Li-O2 Batteries with Organic Carbonate Electrolytes. J. Power Sources 2011, 196, 3894−3899. (14) Freunberger, S. A.; Chen, Y.; Peng, Z.; Griffin, J. M.; Hardwick, L.; Bardé, F.; Novák, P.; Bruce, P. G. Reactions in the Rechargeable Lithium-O2 Battery with Alkyl Carbonate Electrolytes. J. Am. Chem. Soc. 2011, 133, 8040−8047. (15) Laino, T.; Curioni, A. A New Piece in the Puzzle of Lithium/Air Batteries: Computational Study on the Chemical Stability of Propylene Carbonate in the Presence of Lithium Peroxide. Chemistry 2012, 18, 3510−20. (16) Laino, T.; Curioni, A. Chemical Reactivity of Aprotic Electrolytes on a Solid Li2O2 Surface: Screening Solvents for Li-Air Batteries. New J. Phys. 2013, 15, 095009. (17) Read, J. Ether-Based Electrolytes for the Lithium/Oxygen Organic Electrolyte Battery. J. Electrochem. Soc. 2006, 153, A96−A100. (18) Xu, W.; Xiao, J.; Zhang, J.; Wang, D.; Zhang, J.-G. Optimization of Nonaqueous Electrolytes for Primary Lithium/Air Batteries Operated in Ambient Environment. J. Electrochem. Soc. 2009, 156, A773−A779. (19) Kamaya, N.; Homma, K.; Yamakawa, Y.; Hirayama, M.; Kanno, R.; Yonemura, M.; Kamiyama, T.; Kato, Y.; Hama, S.; Kawamoto, K.; Mitsui, A. A Lithium Superionic Conductor. Nat. Mater. 2011, 10, 682−686. (20) Stramare, S.; Thangadurai, V.; Weppner, W. Lithium Lanthanum Titanates: A Review. Chem. Mater. 2003, 15, 3974−3990. (21) Knauth, P. Inorganic Solid Li Ion Conductors: An Overview. Solid State Ionics 2009, 180, 911−916. (22) Kumar, B.; Kumar, J.; Leese, R.; Fellner, J. P.; Rodrigues, S. J.; Abraham, K. M. A Solid-State, Rechargeable, Long Cycle Life LithiumAir Battery. J. Electrochem. Soc. 2010, 157, A50−A54. (23) Quartarone, E.; Mustarelli, P. Electrolytes for Solid-State Lithium Rechargeable Batteries: Recent Advances and Perspectives. Chem. Soc. Rev. 2011, 40, 2525−2540. (24) Anantharamulu, N.; Koteswara Rao, K.; Rambabu, G.; Vijaya Kumar, B.; Radha, V.; Vithal, M. A Wide-Ranging Review on Nasicon Type Materials. J. Mater. Sci. 2011, 46, 2821−2837. (25) Nb, M.; Thangadurai, V.; Kaack, H.; Weppner, W. J. F. Novel Fast Lithium Ion Conduction in Garnet-Type Li5La3M2O12. Adv Energy Mater 2003, 40, 437−440. (26) Thangadurai, V.; Weppner, W. Li6ALa2Ta2O12 (A = Sr, Ba): Novel Garnet-Like Oxides for Fast Lithium Ion Conduction. Adv. Funct. Mater. 2005, 15, 107−112. (27) Thangadurai, V.; Weppner, W. Li6ALa2Nb2O12 (A = Ca, Sr, Ba): A New Class of Fast Lithium Ion Conductors with Garnet-Like Structure. J. Am. Ceram. Soc. 2005, 88, 411−418.

For tetragonal LLZO, our results show clearly that the mechanism of ionic conduction is of synchronous collective nature. It can be interpreted as a consequence of the high ordering within the Li-ion sublattice and the lack of available vacancies. In contrast, for cubic LLZO, we found that a synchronous collective mechanism would result in energetic barriers that are not representative. However, considering an asynchronous mechanism of single-ion jumps that can induce collective motion of a few ions led to good agreement with experimental observations. The large number of vacancies and the disorder in the Li-ion sublattice of cubic LLZO increase the chance of single-ion jumps that can occur at much lower energetic cost than a full synchronous collective motion of several ions at the same time. The single-ion jumps might be understood as activators for ionic conduction in cubic LLZO. The absence of Li-site vacancies in tetragonal LLZO inhibits a comparable mechanism. In summary, the mechanism of synchronous collective motion observed for tetragonal LLZO requires a higher activation energy, which in turn leads to a lower ionic conductivity, than the asynchronous mechanism of single-ion jumps and induced collective motion observed in cubic LLZO. Therefore, the presence of vacancies in cubic LLZO is crucial in order to lower the activation energy and enhance Li-ion conductivity in cubic LLZO. Relying on the outcomes of our work, we speculate that a strategy to design superior Li-ion conductors may involve creation of additional Li-ion vacancies while concurrently stabilizing the cubic phase at room temperature. A way to obtain such goals might be crystal doping using supervalent cations. First experimental approaches show promising results in stabilizing the cubic phase and enhancing Li-ion conductivity in LLZO.34,37,52−54 However, a rationale on the impact of different elemental doping on the stability and on the Li-ion conductivity of cubic LLZO is far from being achieved, and more extensive investigations will be needed.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank the U.S. Department of Energy (DOE) for providing computing time on the IBM® Bluegene®/Q at the ALCF-Argonne National Laboratory (ANL) within the INCITE project on lithium/air batteries and the experimental groups at the IBM Almaden Research Center and at SCHOTT AG (Mainz) for discussion and feedback.

■

REFERENCES

(1) Girishkumar, G.; McCloskey, B.; Luntz, A. C.; Swanson, S.; Wilcke, W. Lithium-Air Battery: Promise and Challenges. J. Phys. Chem. Lett. 2010, 1, 2193−2203. (2) Kraytsberg, A.; Ein-Eli, Y. Review on Li- Air Batteries Opportunities, Limitations and Perspective. J. Power Sources 2011, 196, 886−893. (3) Christensen, J.; Albertus, P.; Sanchez-Carrera, R. S.; Lohmann, T.; Kozinsky, B.; Liedtke, R.; Ahmed, J.; Kojic, A. A Critical Review of Li/Air Batteries. J. Electrochem. Soc. 2012, 159, R1−R30. (4) Park, M.; Sun, H.; Lee, H.; Lee, J.; Cho, J. Lithium-Air Batteries: Survey on the Current Status and Perspectives Towards Automotive 6678



Article

nominal composition Li7La3Zr2O12. Solid State Ionics 2012, 206, 28− 32. (53) Murugan, R.; Ramakumar, S.; Janani, N. High conductive yttrium doped Li7La3Zr2O12 cubic lithium garnet. Electrochem. Commun. 2011, 13, 1373−1375. (54) El Shinawi, H.; Janek, J. Stabilization of Cubic Lithium-Stuffed Garnets of the Type Li7La3Zr2O12 by Addition of Gallium. J. Power Sources 2013, 225, 13−19.

(28) Murugan, R.; Thangadurai, V.; Weppner, W. Fast Lithium Ion Conduction in Garnet-Type Li7La3Zr2O12. Angew. Chem., Int. Ed. 2007, 46, 7778−7781. (29) Awaka, J.; Takashima, A.; Kataoka, K.; Kijima, N.; Idemoto, Y.; Akimoto, J. Crystal Structure of Fast Lithium-Ion-Conducting Cubic Li7La3Zr2O12. Chem. Lett. 2011, 40, 60−62. (30) Awaka, J.; Kijima, N.; Hayakawa, H.; Akimoto, J. Synthesis and Structure Analysis of Tetragonal Li7La3Zr2O12 with the Garnet-Related Type Structure. J. Solid State Chem. 2009, 182, 2046−2052. (31) Kuhn, A.; Narayanan, S.; Spencer, L.; Goward, G.; Thangadurai, V.; Wilkening, M. Li Self-Diffusion in Garnet-Type Li7La3Zr2O12 as Probed Directly by Diffusion-Induced 7Li Spin-Lattice Relaxation NMR Spectroscopy. Phys. Rev. B 2011, 83, 094302−1−094302−11. (32) Geiger, C. A.; Alekseev, E.; Lazic, B.; Fisch, M.; Armbruster, T.; Langner, R.; Fechtelkord, M.; Kim, N.; Pettke, T.; Weppner, W. Crystal Chemistry and Stability of “Li7La3Zr2O12” Garnet: A Fast Lithium-Ion Conductor. Inorg. Chem. 2011, 50, 1089−1097. (33) Buschmann, H.; Dölle, J.; Berendts, S.; Kuhn, A.; Bottke, P.; Wilkening, M.; Heitjans, P.; Senyshyn, A.; Ehrenberg, H.; Lotnyk, A.; Duppel, V.; Kienle, L.; Janek, J. Structure and Dynamics of the Fast Lithium Ion Conductor ‘Li7La3Zr2O12’. Phys. Chem. Chem. Phys. 2011, 13, 19378−19392. (34) Adams, S.; Rao, R. P. Ion Transport and Phase Transition in Li7xLa3(Zr2xMx)O12 (M = Ta5+, Nb5+, x = 0, 0.25). J. Mater. Chem. 2012, 22, 1426−1434. (35) Kasuga, T.; Kanamura, K. Concerted Migration Mechanism in the Li Ion Dynamics of Garnet-Type Li7La3Zr2O12. Chem. Mater. 2013, 25, 425−430. (36) Xu, M.; Park, M. S.; Lee, J. M.; Kim, T. Y.; Park, Y. S.; Ma, E. Mechanisms of Li+ Transport in Garnet-Type Cubic Li3+xLa3M2O12 (M = Te, Nb, Zr). Phys. Rev. B 2012, 85, 052301. (37) Bernstein, N.; Johannes, M. D.; Hoang, K. Origin of the structural phase transition in Li7La3Zr2O12. http://arxiv.org/abs/1209. 0690 2012; 2−6. (38) Cussen, E. J. Structure and Ionic Conductivity in Lithium Garnets. J. Mater. Chem. 2010, 20, 5167−5173. (39) Xie, H.; Alonso, J. A.; Li, Y.; Fern, M. T.; Goodenough, J. B. Lithium Distribution in Aluminum-Free Cubic Li7La3Zr2O12. Chem. Mater. 2011, 3587−3589. (40) Kumar, P. P.; Yashonath, S. Ionic Conduction in the Solid State. J. Chem. Sci. 2006, 118, 135−154. (41) Developers, C. https://www.cp2k.org. 2013; www.cp2k.org. (42) Perdew, J.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (43) Goedecker, S.; Teter, M.; Hutter, J. Separable Dual-Space Gaussian Pseudopotentials. Phys. Rev. B 1996, 1703−1710. (44) Nose, S. A Unified Formulation of the Constant Temperature Molecular Dynamics Methods. J. Chem. Phys. 1984, 81, 511−519. (45) Nosé, S. A Molecular Dynamics Method for Simulations in the Canonical Ensemble. Mol. Phys. 1984, 52, 255−268. (46) Hoover, W. Canonical Dynamics: Equilibrium Phase-Space Distributions. Phys. Rev. A 1985, 31, 1695−1697. (47) Martyna, G.; Tuckerman, M.; Tobias, D.; Klein, M. Explicit Reversible Integrators for Extended Systems Dynamics. Mol. Phys. 1996, 87, 1117−1157. (48) Laio, A.; Parrinello, M. Escaping Free-Energy Minima. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 12562−12566. (49) Henkelman, G.; Uberuaga, B. P.; Jónsson, H. A Climbing Image Nudged Elastic Band Method for Finding Saddle Points and Minimum Energy Paths. J. Chem. Phys. 2000, 113, 9901−9904. (50) Henkelman, G.; Jónsson, H. Improved Tangent Estimate in the Nudged Elastic Band Method for Finding Minimum Energy Paths and Saddle Points. J. Chem. Phys. 2000, 113, 9978−9985. (51) Sakamoto, J.; Rangasamy, E.; Kim, H.; Kim, Y.; Wolfenstine, J. Synthesis of nano-scale fast ion conducting cubic Li7La3Zr2O12. Nanotechnology 2013, 24, 424005. (52) Rangasamy, E.; Wolfenstine, J.; Sakamoto, J. The role of Al and Li concentration on the formation of cubic garnet solid electrolyte of 6679


Solid-State Electrolytes - American Chemical Society

Recommend Documents