Letter pubs.acs.org/NanoLett
Implications of Occupational Disorder on Ion Mobility in Li4Ti5O12 Battery Materials Hendrik H. Heenen,* Christoph Scheurer, and Karsten Reuter Chair for Theoretical Chemistry and Catalysis Research Center, Technische Universität München, Lichtenbergstrasse 4, D-85747 Garching, Germany S Supporting Information *
ABSTRACT: Lithium−titanium-oxide (Li4Ti5O12, LTO) is unique among battery materials due to its exceptional cyclability and high rate capability. This performance is believed to derive at least partly from the occupational disorder introduced via mixed Li/Ti occupancy in the LTO spinel-like structure. We explore the vast configuration space accessible during high-temperature LTO synthesis by Monte Carlo sampling and indeed find lowest-energy structures to be characterized by a high degree of microscopic inhomogeneity. Dynamical simulations in corresponding configurations reveal the dominant fraction of Li ions to be immobile on nanosecond time scales. However, Ti antisite-like defects stabilized by the configurational disorder give rise to a novel correlated ion diffusion mechanism. The resulting fast but localized diffusion could be a key element in the sudden rise in conductivity found in LTO in the early stages of charging and questions the validity of ion mobility measurements for this and other configurationally disordered materials. KEYWORDS: Spinel Li4Ti5O12, Li-ion batteries, diffusion, antisite defects, Wang−Landau, molecular dynamics simulations
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with Ti ions at a ratio of 1:5.9,10 Already in the ideal spinel stoichiometry of LiTi2O4 (Li3Ti6O12), where all 16d sites are filled with Ti, interactions between the tetrahedral Li ions generate a multitude of different local environments in the three-dimensional diffusion network spanned between the 8a sites.11,12 In a material like LTO this configurational space then further explodes due to the fractional occupation of the 16d host lattice sites giving rise to more profound mobility influencing phenomena. Energetically favorable nanoscopic domains with vastly different mobilities may arise just from sampling the possible configurations. For experiments with different inherent length and time scales, like, for example, macroscopic field gradient measurements versus self-diffusion, this nanoscopic and at the same time fluxional inhomogeneity might result in seemingly contradictory views on the true mobility for a material falsely assumed to be ideally homogeneous. This has indeed been recently shown for the case of LTO,13 which makes this commercially applied anode material for lithium ion batteries, valued for its high rate capability, long lifetime, and safe operation,14,15 an ideal object of interest. Sampling the configurational space of LTO with extensive Monte Carlo simulations we show that configurations exhibiting strong variations in the spatial Li and Ti distribution are energetically very close. They are thus all well or easily accessible in the synthesis protocol of Li4Ti5O12, which involves
nderstanding the relation between atomistic lithium hopping mechanisms and macroscopic ion transport kinetics is a key factor in the development of high-performance Li battery electrodes.1−5 Already at the bulk level, this relation goes well beyond the realms of standard dilute diffusion theory. This theory assumes hops of noninteracting Li vacancies or interstitials in an otherwise ideal and homogeneous host lattice, which would then result in a straightforward Arrhenius-type relation between the macroscopic diffusion coefficient and the microscopic activation barrier for individual ionic hops.6 Structural phase transitions in which the host undergoes a crystallographic change upon variation of the Li concentration7 are one of the more pronounced reasons why such a relation holds at best only effectively in real electrode materials then yielding an apparent activation barrier that is void of any direct microscopic meaning. Yet, even at fixed composition and lattice structure interactions among the diffusing charged carriers are known to prevent any simple such Arrhenius-type relation. In the resulting inhomogeneous Li distribution, any migrating ion will sample a variety of different local environments, each leading to a different microscopic activation barrier for the next hop.7,8 Here we argue that yet another complexity level is introduced for novel high-capacity materials exhibiting mixed occupations. Such materials prohibit a clear-cut distinction between mobile Li ions and their host structure, as Li also occupies a certain fraction of lattice sites otherwise filled with host metal cations. In the prototypical spinel-type Li4Ti5O12 (LTO) material, Li, for instance, fills not only all tetrahedral (Td) 8a sites, but also shares the octahedral (Oh) 16d positions © XXXX American Chemical Society
Received: April 4, 2017 Revised: May 16, 2017 Published: May 17, 2017 A
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Figure 1. (left) Configurational density of states g(E) (black line), as well as canonical distributions at 700 K (dark gray area) and 1300 K (light gray area). The energy of the idealized “R3m ̅ ” structure is taken as zero reference (note the discontinuous y-axis necessary to include this high-energy structure). (right) Comparison of the Li16d−Li16d radial distribution functions of the ideal “R3̅m” structure (blue) of the obtained global minimum (yellow) and of a representative configuration from the thermal ensemble (red). The energetic position of these three structures is marked with spheres of corresponding color in the left panel.
high temperatures between TS = 700−1300 K.2,15−20 The varying configurations are then likely frozen-in in the final rapid quench to room temperature. As a result, we expect real mesoscopic LTO electrodes to exhibit a large microscopic inhomogeneity with widely changing local Ti distributions. Depending on this local distribution, we observe the stabilization of Ti antisite-like defects on octahedral 16c sites (Ti16c) in a significant fraction of the thermally accessible configuration space. The defects are surrounded by regions with high correlated Li mobility. This mobility complements vacancy-mediated diffusion mechanisms so far employed as the standard model for solid state diffusion in battery materials.7,8 Having the latter as sole diffusion process is self-contradicting for the early stages of Li intercalation into the Li-depleted Li4Ti5O12. Required 8a vacancies21,22 would face immediate quenching due to a Li surplus, hindering effective transport to the bulk necessary for a phase change to rocksalt type Li7Ti5O12. This conundrum can potentially be resolved by the here found mobility-enhancing Ti16c defects that introduce flexibility to the structure of LTO and act as a seed for longrange diffusion. A picture for LTO follows, which sheds light on the phase-separation during Li insertion23 and rationalizes the sudden conductivity improvement at its early stages.24,25 We explore the configuration space of Li4Ti5O12 through canonical Metropolis Monte Carlo (MC) sampling26 and estimate the relative configurational density of states (DOS) in the relevant total energy range via Wang−Landau sampling.27 Both samplings are based on a density-functional theory (DFT) validated core/shell force field,19,28 as implemented in LAMMPS,29 that we further describe in the Supporting Information (SI). They are carried out in large simulation cells containing (3 × 3 × 3) Fd3̅m bulk unit cells at a total of 1512 atoms according to the chemical formula Li288Ti360O864. In each sampling step, a new configuration is created by interchanging an octahedral Li with a Ti atom, and subsequently performing an optimization of atom positions and cell constants starting from an ideal face-centered cubic (fcc) anion lattice. We initialize the MC sampling from the minimum energy configuration identified in previous firstprinciples studies within the smallest stoichiometry-fulfilling R3̅m space group representation of LTO.21,30 Hereby we periodically repeat this (stoichiometrically) minimal Li8Ti10O24 cell resulting in a highly ordered configuration (denoted as “R3̅m” in the following). Ultrahigh temperatures (700, 1800, and 4200 K) are used to globally sample the corrugated potential energy landscape with a varying degree of occupational disorder. We use the resulting coarse information on the configuration space, obtained from more than 900 000
generated trial structures as the basis for a refining Wang− Landau sampling to finally arrive at the approximate, relative configurational DOS g(E) (see the SI for further details). Via the latter we estimate the canonical distributions, P(E,TS) = g(E)e−E/kBTS with kB as the Boltzmann constant at temperatures TS = 700 and 1300 K.27 These distributions contain the ensemble of microscopic configurational motifs we can expect to be generated by the high-temperature LTO synthesis protocol. The obtained relative DOS g(E) shown in Figure 1 reveals a rapidly rising number of configurations at increasing energies. This confirms the expected richness of the LTO phase space. Intriguingly, the sampling thereby identifies a large number of configurations at energies well below the ideal reference 21,30 structure “R3m In fact, the ̅ ” focused on in previous work. thermal ensembles at 700 and 1300 K represented by the canonical distributions in Figure 1 are both even peaked at energies much lower than “R3̅m”. The artificial periodicity enforced by the limited system sizes tractable with firstprinciples calculations thus yields a high-energy structure that is not at all representative of the LTO phase space accessible during synthesis. Even though contained within a narrow energy range of a few meV per atom, the lower-energy structures within the thermal ensembles exhibit a largely varying degree of order with respect to the distribution of Li and Ti atoms on the octahedral 16d sites. We illustrate this structural inhomogeneity in Figure 1 by contrasting the Li16d− Li16d radial distribution functions (RDFs) of the ideal “R3̅m” structure (blue) of the lowest energy structure identified in our sampling (yellow) and of a structure taken from the peak of the 1300 K ensemble (red). While the first exhibits an even RDF characteristic of the ordered Li16d distribution, the latter two show significantly more uneven RDFs including satellite peaks indicative of a largely disordered Li arrangement already on the scale of our microscopic simulation cells. As a result of the hightemperature synthesis protocol, a mesoscopic LTO electrode will correspondingly feature a rather distinct intrinsic inhomogeneity with widely changing local Li and Ti distributions. Intriguingly, we find this local occupational disorder to stabilize antisite-like defects. They are characterized by a displacement of a Ti ion from a 16d to a 16c site and a shift of the Li ion on an adjacent 8a to the original 16d site (see Figure 2). Because of the spatial constriction of a connecting tetrahedral 48f site, relaxation of this Ti 16c occupation back to the regular 16d site is kinetically hindered at ambient temperatures. Nudged-elastic-band (NEB) calculations31,32 performed for corresponding antisite defects in the obtained B
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Figure 2. (left) Crystallographic arrangement of the Ti16c antisite defect with arrows indicating the displacement of the Ti (curved arrow) and Li (linear arrow) ion generating the defect. (right) Schematic illustration of a concerted interstitialcy diffusion process enabled in the vicinity of the Ti16c antisite defect (gold). Shown is the frequent back-and-forth hop of one Li ion to the metastable 16c site (light blue spheres). The thereby created short-lived 8a vacancy enables the true diffusive motion of a second Li ion (dark blue sphere). Oxygen ions are shown in red, Ti ions in gray, and Li ions in green.
crucial role for the true diffusion in locally disordered regions around Ti16c antisite defects. The obtained MD data reveals a clear correlation between true Li ion mobility and the antisite defect. Configurations not exhibiting this defect do not show any Li ion diffusion on the nanosecond time scale. This extends to configurations at both low and high energies of the thermal ensembles identified in Figure 1 and holds thus irrespective of the wide range of local disorder otherwise probed. This is consistent with the wellknown low ionic conductivity of stoichiometric LTO14,39 and confirms recent ensemble-probing NMR results.24,25 In contrast, configurations containing a Ti16c antisite defect show a small share of Li ions to be mobile among a dominant share of immobile Li ions. These mobile Li ions nevertheless remain in a finite region of about 10 Å around the Ti16c defect, implying localized diffusion. Among the mobile Li ions, a portion performs only frequent back-and-forth jumps to the metastable intermediate site (dTd−Oh ≤ d < 2dTd‑Oh). True diffusive motion (d ≥ 2dTd−Oh) arises instead in a concerted process. The short-lived vacancies created by the evasive hops into the metastable site enable a rare consecutive multiple-site migration by up to three secondary Li ions as illustrated in Figure 2. As such the process largely resembles the interstitialcy mechanism often found in fast ion conductors.6 This is intriguing as it reveals that LTO does exhibit the materials’ properties desirable for a good ion conductor. Because of the configurational disorder it just exhibits them only locally in nonpercolating regions. NEB calculations detailed in the SI indeed indicate minimum energy barriers in the range 0.1−0.2 eV for this kind of interstitialcy process, which rationalizes its observation in our MD simulations. In contrast, embedding the same mechanism within regions not in proximity of a Ti16c antisite defect yields NEB barriers in excess of 0.8 eV, underscoring the close connection between the diffusion mechanism and the defect. The picture thus emerging from our study shows stoichiometric, nominally homogeneous LTO as a mesoscopic material composed of largely varying, but predominantly rigid local configurations that exhibit little Li ion mobility. This is interspersed with a lower bound of 50−100 ppm of Ti16c antisite defects per Ti atom, which generate regions showing localized, correlated diffusion. The extent of the latter can be estimated from the spatial occupation of mobile lithium ions during the MD trajectories and the overall defect concentration. As a likely largely underestimated value we arrive at 0.01−0.2% of the total materials volume (see above and SI). The correlated diffusion complements prevalent vacancy-mediated diffusion and is fundamentally different in its mechanism. In vacancy-
thermal ensembles yield barriers in excess of 2.3 eV for this process, cf. SI. A rough estimate for the defect formation probability can be obtained by introducing and optimizing single such defects into randomly chosen configurations from the structural ensemble, cf. SI. On the basis of this we estimate the generated concentration of these defects at the elevated synthesis temperatures to be around 50−100 ppm per Ti ion. This number is likely underestimated because we found the employed force field to occasionally overestimate the defect formation energies in comparison to DFT and we have confined the defect density to one defect per simulation cell. Additionally, the estimate represents the theoretical limit of the thermodynamic equilibrium that can easily underscore experimental findings by an order of magnitude, as, for example, found for antisite defects in lithium−iron-phosphate.33 The here obtained concentration would be at least 2 orders of magnitude below the sensitivity of X-ray diffraction measurements16,34 but is nevertheless significant in terms of Li ion mobility as shown in the following. We demonstrate the impact of the occupational disorder and the antisite defects on the local Li ion diffusion with molecular dynamics (MD) simulations carried out at 300 K for a number of configurations selected from the sampled thermal distributions. For each configuration, seven MD runs of 1.7 ns each are conducted in the NVE ensemble after initial equilibration by a Nose−Hoover chain thermostat and a Hoover chain barostat (cf. SI for details).35,36 To analyze the Li ion mobility from the simulated trajectories we focus in the following on the maximum displacement d that each individual Li ion reaches from its original position at any time during the simulation. By comparing d to the distance dTd−Oh ∼ 1.8 Å between a neighboring tetrahedral and octahedral site in the ideal LTO lattice, we distinguish three mobility regimes: d < dTd−Oh reflects immobile Li ions, dTd−Oh ≤ d < 2dTd−Oh indicates at maximum a movement into a neighboring coordination site, and d ≥ 2dTd−Oh reflects migration between multiple (more than one) crystallographic sites. In the SI, we compare this to a complementary analysis that employs a discretization procedure similar to methods presented in literature37,38 which allows for the evaluation of the total number of jumps between different sites during the entire MD trajectory. This analysis reveals that indeed only the multiple-site migration corresponds to a true diffusive pathway, for instance via sites 8a ↔16c ↔8a. The intermediate displacements dTd−Oh ≤ d < 2dTd−Oh instead predominantly arise from jumps into a metastable intermediate state (mostly 8a →16c), from which the ion returns on a picosecond time scale. This intermediate state was already identified in previous work7,21,22,39 and we show below its C
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scales occurring in these materials at the same time. They thus each constitute an oversimplification. The full picture is richer and definitely more interesting.
mediated diffusion, Li mobility arises from a local understoichiometry. Considering ion transport in spinel-LTO, such an under-stoichiometric vacancy mechanism alone does not provide a realistic homogeneous bulk scenario, as Li concentration in Li4Ti5O12 is already at a minimum with respect to the battery charge state. This vacancy-devoid Li4Ti5O12 is the stoichiometric state that also forms the basis for our simulations. Loading Li4+xTi5O12 with Li ions (up to x = 3) is known to proceed via the formation of domains of a rock-salt structured phase (Li7 Ti4O 12 ) even distinguishable within primary particles.14,23 At the (highly dynamic) interface between spinel and rock-salt domains, Li vacancies form locally during the interface migration.40 These local, individual Li vacancies are likely to trigger the correlated motion of multiple Li ions along the percolation segments around the Ti16c antisite defects observed in our simulations. This drains long-range connected channels toward the phase boundary and creates further vacancies at the remote end of the segment. These are significantly displaced from the original vacancy due to the finite segment length (here found at a minimum of 8−12 Å) presenting an effective long-range vacancy transport into the bulk. This cascade-like behavior would thus result in the phase boundary acting as a natural attractor in the vicinity of a defect, funneling Li ions toward the Li-rich phase when collecting percolation segments during the phase growth. Vice versa, such directed paths could seed rock-salt domains embedded in the spinel-phase. The proposed mechanism could thereby explain percolation channels observed during lithiation of LTO through conductive atomic force microscopy experiments.41 Additionally the cascade-like mechanism likely leads to enhanced mobility of even dilute vacancies that elucidates the sudden surge in conductivity found in LTO on minute (x = 0.1) lithiation of Li4+xTi5O12 mimicking the early stages of charging.24,25 On a more general note, we expect similar mechanisms leading to profound differences in local ion mobility to be rather abundant for high-performance battery materials and superionic conductors. Many prominent materials (e.g., Li7La3Zr2O12 (LLZO)42 and Li3xLa2/3−xTiO3 (LLTO)43) with proven high ionic conductivities are characterized by a high degree of structural and occupational flexibility, thus opening up vast configurational spaces by local disorder. As we have shown, any picture of ion mobility derived from an averaged view and projected onto small, idealized unit cells must thus fail in two ways: (i) in the energetics of the most stable structures, where locally disordered motifs accommodate each other and facilitate the formation of defects, and (ii) in the microscopic ion dynamics, which by transition state theory depends with exponential sensitivity on the wide spectrum of local surroundings. The resulting distribution of a plethora of nanoscopic domains with high mobility embedded into an inhomogeneous background presents a structural picture which is neither solid solution nor separate phases, but instead a truly nanostructured material. That this finding is not only of academic theoretical relevance is underpinned by the fact that it naturally rationalizes apparent discrepancies in experimental observations, such as between SLR-Li NMR and impedance spectroscopy of LTO. These show several different apparent activation energies for ion transport,39 none of which are clearly reflecting the materials high rate capabilities.44 The homogeneous average models employed in each experimental field’s standard data analyses just can not match all length and time
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.7b01400. Full details to the validation of the employed force field, sampling methods, nudged-elastic-band simulations, molecular dynamics simulations, and discretization procedure (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Hendrik H. Heenen: 0000-0003-0696-8445 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Funding by the International Center for Energy Research (ICER), an international cooperation between Technical University Munich (TUM) and Nanyang Technological University (NTU), Singapore, is gratefully acknowledged. Computational resources have been provided through a research project granted by the Jülich Supercomputing Centre (JSC) at the Jülich Research on Exascale Cluster Architectures (JURECA) HPC system.
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