# Sparse cyclic excitations explain the low ionic conductivity of stoichiometric Li7La3Zr2O12

Mario Burbano, Dany Carlier, Florent Boucher, Benjamin J. Morgan,
Mathieu Salanne

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Mario Burbano, Dany Carlier, Florent Boucher, Benjamin J. Morgan, Mathieu Salanne. Sparse cyclic excitations explain the low ionic conductivity of stoichiometric Li7La3Zr2O12. Physical Review Letters, 2016, 116 (13), 135901 (6 p.). 10.1103/PhysRevLett.116.135901. hal-01300141

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Sparse Cyclic Excitations Explain the Low Ionic Conductivity of Stoichiometric
Li<sub>7</sub>La<sub>3</sub>Zr<sub>2</sub>O<sub>12</sub>

Mario Burbano<sup>1,2,3</sup>, Dany Carlier<sup>2,3</sup>, Florent Boucher<sup>4</sup>, Benjamin J.Morgan<sup>5</sup>, and Mathieu Salanne<sup>1,3</sup>

<sup>1</sup>Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 8234, PHENIX, F-75005 Paris, France.

<sup>2</sup>CNRS, Université de Bordeaux, ICMCB, 87 avenue du Dr A. Schweitzer, 33608 Pessac Cedex, France.

<sup>3</sup>Réseau sur le Stockage Electrochimique de l'Energie (RS2E), FR CNRS 3459, France

<sup>4</sup>Institut des Matériaux Jean Rouxel (IMN), Université de Nantes, CNRS,
2 rue de la Houssinière, BP 32229, 44322 Nantes cedex 3, France. and

<sup>5</sup>Department of Chemistry, University of Bath, Claverton Down BA2 7AY, United Kingdom*

(Dated: November 5, 2018)

We have performed long time-scale molecular dynamics simulations of the cubic and tetragonal phases of the solid lithium-ion-electrolyte Li<sub>7</sub>La<sub>3</sub>Zr<sub>2</sub>O<sub>12</sub> (LLZO), using a first-principles param- eterised interatomic potential. Collective lithium transport was analysed by identifying dynami- cal excitations; persistent ion displacements over distances comparable to the separation between lithium sites, and string-like clusters of ions that undergo cooperative motion. We find that dynami- cal excitations in c-LLZO are frequent, with participating lithium numbers following an exponential distribution, mirroring the dynamics of fragile glasses. In contrast, excitations in t-LLZO are both temporally and spatially sparse, consisting preferentially of highly concerted lithium motion around closed loops. This qualitative difference is explained as a consequence of lithium ordering in t- LLZO, and provides a mechanistic basis for the much lower ionic conductivity of t-LLZO compared to c-LLZO.

Conventional lithium-ion batteries rely on unstable liquid-organic polymer electrolytes, which pose practical limitations in terms of flammability, miniaturization, and safe disposal. A possible solution is to replace liquid elec- trolytes with inorganic ceramics that are electrochemi- cally stable and non-flammable. The family of garnet- like oxides with general formula Li<sub>x</sub>M<sub>3</sub>M'<sub>2</sub>O<sub>12</sub>, where M = La and M' = Nb, Ta or Zr, have attracted signifi- cant attention in this regard due to their high lithium- ion conductivity, high electrochemical stability window, and chemical stability with respect to metallic lithium [1, 2]. The highly stuffed garnet Li<sub>7</sub>La<sub>3</sub>Zr<sub>2</sub>O<sub>12</sub> (LLZO) is the most studied member of this family, and can be considered prototypical. LLZO exhibits two phases with strikingly different ionic conductivities: a cubic phase (c- LLZO) that is adopted at high temperature (> 600 K) or stabilized by doping [3, 4] with σ ≈ 10<sup>-4</sup> S cm<sup>-1</sup>, and a tetragonal (t-LLZO) phase with σ ≈ 10<sup>-6</sup> S cm<sup>-1</sup> that is favoured in the pure system at ambient temperature.

The large difference in ionic conductivity between c-
LLZO and t-LLZO is interesting from a mechanistic
perspective because the pathways available for lithium
transport are topologically identical in the two phases.
Lithium ions move through an open three dimensional
network of rings. Each ring comprises twelve alternat-
ing tetrahedral and octahedral sites (Fig. 1) [5], and the
tetrahedral sites act as nodes connecting adjacent rings.
In stoichiometric LLZO each ring accommodates on av-
erage eight lithium ions, which preferentially occupy all
six octahedra and two of the tetrahedra. In the cubic
phase the tetrahedral sites are equivalent and the lithium
ions are disordered. In the tetragonal phase the tetrahe-
dra are inequivalent due to the reduced crystal symme-
try, and lithium occupies tetrahedral pairs aligned along

the [001] direction forming an ordered sublattice [3, 4].
This lithium ordering is correlated with the sharp de-
crease in ionic conductivity relative to the cubic phase.
A mechanistic explanation of the relationship between
lattice symmetry, lithium ordering, and ionic transport
is, however, lacking.

Previous theoretical studies identified that lithium dif-
fusion in c-LLZO and t-LLZO phases occurs via corre-
lated ionic motion [6–8]. The relationship between mi-
croscopic ion-correlations and macroscopic lithium trans-
port, characterised by diffusion coefficients and ionic con-
ductivities, is however unresolved. Meier *et al*. used *ab
initio* molecular dynamics (MD) and metadynamics sim-

<figure><img src="image_2.png" /><figcaption>FIG. 1. Schematic of the polyhedral ring topologies in (a) cubic and (b) tetragonal LLZO. Triangles represent tetrahedra, and rectangles represent octahedra. Arrows indicate neighbouring octahedra within conjoined rings. In c-LLZO lithium resides at off-center positions in the octahedra, and is disordered over all available octahedra and tetrahedra. In t-LLZO lithium is ordered over two tetrahedra and four octahedra in each ring. White circles show these lithium sites, which are partially occupied in c-LLZO.</figcaption></figure>
ulations to study cubic and tetragonal LLZO [8]. These authors observed diffusion in c-LLZO effected by asyn- chronous correlated hopping of lithium ions, while in t- LLZO a synchronous concerted process involving multi- ple lithium ions was identified. In contrast, an earlier ab initio MD study of c-LLZO by Jalem et al. attributed diffusion in the cubic phase to synchronous lithium hop- ping [7]. The apparently inconsistent descriptions from these studies illustates the challenge of adequately sam- pling ionic transport in materials with complex correlated diffusion mechanisms. Direct first principles simulations are computationally costly, and these two studies con- sidered relatively short simulation trajectories (10–30 ps) that captured only a few diffusion events [9]. This limited sampling means it is not clear whether individual dif- fusion processes represent long-time ensemble diffusion behaviour. If we wish to connect microscopic lithium dynamics with macroscopic transport behaviour, much longer simulations and a statistical treatment of the re- sulting trajectories are necessary.

Here we report long time-scale classical MD simula-
tions of cubic and tetragonal LLZO, with particular em-
phasis given to analysing their room temperature lithium
transport. These simulations use an interatomic poten-
tial (IP) model for lithium garnets derived from state-
of-the-art hybrid Density Functional Theory (h-DFT)
calculations. This allows much longer simulation times
than direct ab initio calculations at moderate compu-
tational expense, while avoiding empirical parameterisa-
tion [10]. We have characterised lithium transport in
both phases using two techniques typically used to study
glass-forming liquids [11, 12], and not previously applied
to solid electrolytes. This analysis reveals qualitatively
different transport statistics for c-LLZO versus t-LLZO.
We identify displacive excitations, defined as persistent
ion displacements over distances comparable to the sep-
aration between lithium sites. Excitation events are fre-
quent in c-LLZO, but highly sparse in t-LLZO. A statis-
tical comparison of lithium diffusion behaviour is made
by identifying string-like clusters of ions that undergo
cooperative motions [12]. In c-LLZO the probability dis-
tribution of string lengths is exponential, mirroring the
behaviour of fragile glassy liquids [12, 13]. In contrast,
the string length distribution in t-LLZO is discontinuous,
with distinct preferred string lengths. The most probable
string lengths correspond to cyclic cooperative motions of
lithium ions around closed loops. These cyclic processes
give zero net displacement of charge, and therefore do not
contribute to ionic conductivity. The low conductivity of
t-LLZO is therefore attributed to persistent diffusive pro-
cesses being sparse in location and time, and dominated
by zero-charge displacement closed-loop cyclic processes.

Classical MD allow simulations across the length- and
timescales necessary to describe collective ionic motion
at a fraction of the computational cost of ab initio meth-
ods. This efficiency often comes at the expense of ac-

currency. One solution is to use physically motivated IPs
that reproduce the electron density response of individ-
ual ions to changes in their coordination environment
[14]. Parameters for the resulting models can be derived
from electronic structure (i.e. DFT) calculations, to pro-
duce IPs that are accurate and transferable [15]. Such
IPs are capable of describing changes in composition and
local structure, such as those which occur during ionic
transport, or across families of similar materials. Param-
eters for a DIPole Polarizable Ion Model (DIPPIM) [14],
used throughout this work, were obtained by calculating
sets of forces, dipole moments, and stresses using h-DFT
across a sample of different stoichiometries and atomic
geometries, and then minimising the least squares er-
rors between the h-DFT and DIPPIM data. Although
computationally expensive, h-DFT functionals give im-
proved structural parameters, such as lattice constants,
compared to standard DFT [15, 16]. A full description
of the parameterization procedure and validation of the
resulting IP against experimental structural data is given
in the Supplementary Information.

We performed molecular dynamics simulations of the
lithium garnet Li₇La₃Zr₂O₁₂, using a 2 × 2 × 2 supercell
containing 1536 atoms, with a timestep of 1 fs. The sys-
tem was equilibrated for temperatures ranging between
300 K and 1000 K in the isothermal-isobaric (NPT) en-
semble. The supercells were initially equilibrated at a
temperature of 280 K for 10 ps; the temperature was then
scaled up to 1000 K at a rate of 1 K ps⁻¹. Production
runs were performed in the canonical (NVT) ensemble
using the equilibrium values from the NPT simulations
and were up to 87.4 ns long, in the case of t-LLZO at
300 K.

Fig. 2 shows lithium mean-squared displacements cal-
culated at temperatures from 300 to 1000 K [17]. At
high temperatures, where the c-LLZO phase is adopted,
lithium diffusion is fast and the diffusive regime is sam-
pled even by short simulation trajectories. In contrast,
below 600 K, where the tetragonal phase is favoured, the
msd is orders of magnitude smaller. To sample the diffu-
sive regime well, long-time simulations become necessary,
which would be prohibitively costly for ab initio meth-
ods.

To characterise lithium transport in materials with
concerted ionic diffusion, one cannot simply examine
hopping frequencies of individual ions (as for simple in-
terstitial or vacancy diffusion), but must instead consider
the collective motions of groups of atoms and the contri-
butions made by these to ensemble transport coefficients
[18]. Providing that a system is in a particle hopping
regime (i.e. is not “superionic” [19]) local diffusion events
are well separated in time. Net contributions to ensem-
ble transport correspond to non-trivial particle displace-
ments and exclude vibrational motion or very short lived
configurational changes. We identify transitions between
relatively long-lived configurations, or “excitations”, us-
<figure><img src="image_3.png" /><figcaption>FIG. 2. Calculated mean-squared displacements for LLZO at 300–1000 K. On a log–log plot, the diffusive regime corresponds to the sloping linear portion.</figcaption></figure>

ing the methodology of Keys et al. for supercooled glass forming liquids [11]. In this case, mobile ions involved in excitations are selected using the following functional of particle trajectories:

$$h_i(t, t_a; a) = \prod_{t'=t_a/2-\Delta t}^{t_a/2} \theta(|\mathbf{r}_i(t+t') - \mathbf{r}_i(t-t')| - a), \quad (1)$$

where a is a displacement cutoff, Δt is the typical time for a mobile particle to move to a distinct position, here 5 ps, and $t_a > \Delta t$ is a sufficiently long time to allow a complete transition between microstates, here set to 30 ps. θ is the Heaviside step function, θ(x) = 1 or 0 for x ≥ 0 or < 0, respectively. The product is evaluated for each lithium ion at every frame of the simulation trajectory. Summing over all particles each frame then gives the number of particles involved in each excitation event. For our analysis of both LLZO phases, a = 3 bohr, which is approximately the Li–Li intersite distance.

To allow a direct comparison between phases, a c-LLZO cell was also simulated at 300 K by enforcing a cubic lattice shape. Excitation statistics for c-LLZO and t-LLZO at 300 K are shown in Fig. 3. Each peak corresponds to an excitation event, where the peak height is the number of participating lithium ions. For c-LLZO, even at 300 K there are many excitations, with well-distributed numbers of contributing atoms. The t-LLZO excitation statistics, however, are strikingly different. Even at the long simulation timescale considered (87.4 ns), few excitations are observed. Collections of ions move in close succession, and many nanoseconds pass between events with no persistent diffusive motion. The t-LLZO data also show a strong preference for specific excitation sizes. Ten of the twelve observed excitations involve exactly eight lithium ions, suggesting a specific eight-atom process dominates lithium dynamics.

<figure><img src="image_4.png" /><figcaption>FIG. 3. Excitation statistics at 300 K for cubic (upper panel) and tetragonal (lower panel) LLZO.</figcaption></figure>

As a supporting analysis of the lithium diffusion statistics in c-LLZO and t-LLZO, we identify “strings” of lithium involved in cooperative diffusion, using a procedure proposed by Donati et al. for studying dynamics in supercooled liquids [12]. The original procedure generates strings by connecting pairs of mobile particles i and j if

$$\min (|\mathbf{r}_i(t) - \mathbf{r}_j(0)|, |\mathbf{r}_j(t) - \mathbf{r}_i(0)|) < r_{\min}. \quad (2)$$

In a solid-lithium electrolyte, such as LLZO, lithium ions move between well defined coordination tetrahedra and octahedra. Applying this concept of lithium hopping between lattice sites allows a modified connection criterion of

$$S_i(t + \Delta t) = S_j(t), \quad (3)$$

where $S_i(t)$ is the site occupied by ion i at time t. The set of oxygen coordinates at each time step define instantaneous polyhedra geometries, which allow us to assign every lithium to a specific lattice site. To reduce noise from short-lived vibrational motions (which do not contribute to diffusion) we filter changes in site occupation for each lithium that are reversed at the next time step.

Fig. 4 shows the probability distributions, $P(n_{string})$, that a diffusing Li ion contributing to a string of length $n_{string}$. Echoing the differences in excitation statistics, the string statistics for the two phases are qualitatively different. For c-LLZO strings of all lengths are found, and $P(n_{string})$ is a monotonically decreasing exponential distribution, as seen for supercooled glassy liquids [12, 13]. For t-LLZO, however, strings of length $n_{string} > 1$ are, in general, much less likely, except for specific values of $n_{string}$. The most probable connected string length
<figure><img src="image_5.png" /><figcaption>FIG. 4. Probability distribution, $P(n)$ of string lengths $n$ for c-LLZO (upper) and t-LLZO (lower). $\Delta t = 30$ ps, for consistency with the excitation analysis.</figcaption></figure>

is $n = 8$; nearly two orders of magnitude more frequent than $n = \{7, 9\}$, and an order of magnitude more frequent even than $n = 2$.

Our analysis reveals a preference in t-LLZO for diffusion events involving eight lithium ions, that does not exist in c-LLZO. Directly examining the trajectories of ions in one of these eight-lithium diffusion events we find cooperative cyclic processes, with eight lithium ions moving around a twelve-site ring. One such cyclic excitation is illustrated in Fig. 5, which shows the initial positions of the eight contributing lithium ions alongside their trajectories and individual displacements. Longer processes that involve more lithium ions are also found. The t-LLZO excitation statistics reveal a single excitation with $n = 14$, which matches the second peak in $P(n_{\text{string}}) = 14$. These 14 lithium processes are also closed loops, that involve coherent lithium motion around two rings. Another excitation involving 27 ions is also observed, which extends across the simulation cell boundary before meeting with its periodic image (Fig. S2).

The strong preference for eight-lithium cyclic diffusion in t-LLZO can be understood as a consequence of the tetragonal lattice symmetry and associated lithium ordering [3]. The stoichiometry of LLZO gives exactly eight lithium ions per ring. Each ring has six octahedral and six tetrahedral sites. In t-LLZO the lithium preferentially occupies all six octahedral sites and two of the tetrahedral sites on opposite sides of the ring [5]. The tetrahedral sites are nodes connecting rings and the orientation of occupied tetrahedral sites in one ring is therefore correlated with its neighbours, producing a fully ordered phase (ignoring thermal disorder) [3]. In an ordered phase an arbitrary set of mobile ions moving between sites may

<figure><img src="image_6.png" /><figcaption>FIG. 5. An example 8-ion cyclic excitation event in t-LLZO. The parameters used in the excitation functional, Eqn. 1, are also shown.</figcaption></figure>

be classified according to whether the initial and final configurations are (locally) equivalent by symmetry. We distinguish complete processes, where the start and end points are equivalent, from partial diffusion processes. In t-LLZO, a complete diffusion process conserves the arrangement of occupied tetrahedral sites. From this perspective, the eight-lithium cyclic excitations can be explained as the shortest possible complete diffusion process. Longer complete processes are also possible—for example, concerted diffusion of 14 lithium ions around a closed loop of two rings—but less likely.

The much lower ionic conductivity of t-LLZO compared to c-LLZO can therefore be attributed to two factors. First diffusive excitations in t-LLZO are highly sparse, and occur much less frequently than in c-LLZO. Secondly, excitations in t-LLZO are predominantly isolated cyclic processes around closed loops. These have zero net charge displacement, and therefore do not contribute to ionic conductivity. If only closed cyclic diffusion processes occurred in t-LLZO the ionic conductivity would be zero. We can speculate that in a macroscopic system occasional excitations will meet and coalesce with a second excitation, then a third, and on, thus allowing ionic conduction. The 27 ion excitation we observe shows an analogue of this behaviour. This excitation crosses a cell boundary and connects with itself because of the simulation periodic boundary conditions.

In conclusion, we have performed long-time MD simulations of cubic and tetragonal LLZO using a first principles based IP. Our long simulation times allow thorough sampling of the lithium diffusion behaviour, even in the poorly conducting t-LLZO phase. By applying statistical analyses, previously used to study transport in glassy materials, we find lithium diffusion in c-LLZO and t-LLZO is qualitatively different. In c-LLZO persistent dynamical excitations are frequent and occur with exponentially distributed numbers of participating lithium ions, showing behaviour characteristic of supercooled glasses [12, 13]. In t-LLZO persistent excitations are sparse, rare events,
and are dominated by concerted ionic motion around the rings that make up the lithium polyhedral network. The preference for closed loop diffusion is due to the strong lithium ordering in the tetragonal phase. The low ionic conductivity of t-LLZO compared to c-LLZO is therefore explained as a consequence of the dominant diffusive excitations in t-LLZO being sparse cyclic processes.

More generally, these results illustrate the importance of thorough statistical sampling, via long simulation trajectories, when modelling solid electrolytes that exhibit concerted ionic motion. We have shown the utility of first-principles-parameterised, transferable IPs in accurately modelling electrolyte materials over long simulation times, and how analysis techniques commonly used for studying glassy liquids can resolve statistically different diffusion behaviour in solid electrolytes.

This work was supported by the French National Research Agency (Labex STORE-EX: grant ANR-10-LABX-0076). We thank the University of Bordeaux for the use of the cluster Avakas. We are grateful for the computing resources on OCCIGEN (CINES) obtained through the project x2015097321. B. J. M. acknowledges support from the Royal Society (UF130329).

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