Transport in Frustrated and Disordered SolidElectrolytes

Boris Kozinsky

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Transport Mechanism Analysis and Coarse-Graining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Correlation and Collective Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Geometric Sublattice Frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.1 Algebraic View of Sublattice Frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Extrinsic Disorder: Amorphous Materials and Random Alloys . . . . . . . . . . . . . . . . . . . . . . 17

6.1 Host Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Abstract

Materials design for next-generation solid-state Li-ion batteries requireatomistic-level understanding of ionic transport mechanisms. It is not wellunderstood what the required ingredients and design principles are for enablinghigh ionic conductivity of inorganic materials at room temperature. Strong ion-ion interaction, geometric frustration, disorder, and collective motion seem to beemerging as common themes in recent investigations of super-ionic materials.This chapter presents an overview of the essential features of these effects, theirrelationships, and implications on ionic transport. We illustrate applicationsof atomistic computational tools in combination with techniques of statistical

B. Kozinsky (�)John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge,MA, USAe-mail: bkoz@seas.harvard.edu

© Springer International Publishing AG, part of Springer Nature 2018W. Andreoni, S. Yip (eds.), Handbook of Materials Modeling,https://doi.org/10.1007/978-3-319-50257-1_54-1

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2 B. Kozinsky

mechanics, group theory, and computational geometry for analyzing dynamicsand transport mechanisms in the context of understanding and optimization ofleading oxide Li-ion conductors.

1 Introduction

Ability of complex oxide materials to conduct ions lies at the heart of key moderntechnologies with high technological impact. Solid-state batteries, chemical sensors,solid oxide fuel cells, gas separation membranes, and memristors rely on ionictransport for their function. In particular, safe high-energy batteries for automotiveand stationary energy storage can unlock a market of several hundred billiondollars, making it arguably the largest technology impact enabled by new functionalmaterials. Today’s battery electrolytes are based on liquid organic solvents thatare combustible and prone to decompose and passivate the interfaces with elec-trodes, causing performance degradation. It would be a major technological stepforward to replace the liquid electrolyte in a battery with a solid ion conductorthat is chemically and mechanically stable against the Li metal anode, whosehigh gravimetric capacity can double the energy density of conventional cellswith graphite anodes. Over the past two decades, significant progress has beenachieved in optimizing solid-state Li-ion conductors. However, only a few classesof fast Li-ion conductors are known, and they have been discovered and optimizedprimarily by haphazard trial and error. Highest ionic conductivities are found insulfides, particularly crystalline Li10GeP2S12 (12 mS/cm at room temperature) andglassy xLi2S-yP2S5 (3 mS/cm). Oxides are more chemically and electrochemicallystable but have lower conductivity, for instance garnets Li6.55L3Zr2Ga0.15O12 withconductivity of 1.6 mS/cm and NASICON Li1.3Al0.3Ti1.7(PO4)3 with 1 mS/cm(Meesala et al. 2017). The great technological importance but limited numberof fast ion conductors has inspired recent efforts to identify new families ofmaterials more suitable for application in solid-state batteries, but this is inherentlychallenging given the nonlinear dependence of conductivity on atomic structureand composition. Accelerated materials development requires identification ofgeneral design rules and mechanistic principles enabling fast ion transport. Variousqualitative descriptions have been proposed over the last decade, such as sufficientvolume of conduction pathways and high anion polarizability, but these have limitedpredictive power. In order to make rational progress in the search and optimizationof fast ion conductors, it is important to develop general physics-based modelsand validate their prediction using statistically significant data sets. Formulating ageneral microscopic theory of super-ionic transport in solids is difficult because ion-ion interactions and correlations are especially strong when carrier concentration ishigh. Even qualitative mechanistic principles and design rules that govern “super-ionic” behavior are not yet established, and generalization from only a few knowncases is difficult. This is in stark contrast with the success of band theory andFermi liquid model in perturbatively describing electron interactions and transportin weakly correlated semiconductors and metals. There is evidence that solid-state

Transport in Frustrated and Disordered Solid Electrolytes 3

ionic transport in the best conductors exhibits collective features reminiscent ofcorrelated fluids, where ionic transport is typically much faster than in regular solids.Despite the long history of the field, complex effects of many-body interactionson transport in atomic fluids are generally not well captured by analytical models,making explicit atomistic simulations particularly important. In this chapter, weillustrate how combinations of techniques of statistical mechanics, group theory, andmolecular dynamics can help develop systematic understanding of ionic transportin strongly interacting and dynamic systems. We give an overview of practicalcomputational approaches useful for analyzing transport mechanisms, studyinggeometric frustration, and estimating correlation in crystalline and disorderedmaterials and provide examples of their application to understand solid oxide Li-ion electrolytes.

2 Simulation Methods

Fast ion conduction arises due to a delicate balance between the interatomic forcesacting on the mobile ions. On one hand, the mobile carrier sublattice needs to besufficiently coupled to the host lattice in order to ensure overall thermodynamic andmechanical stability of the material. On the other hand, the interaction of the mobileions with their host cannot be too strong so as to allow for low enough activationbarriers in order for ionic current to flow at device operating temperatures. Thestrong ionic interactions of Li ions with the host lattice determine the topology ofsites that can be occupied, typically triangles, tetrahedra, or octahedra formed bythe anions (e.g., O, S, N, or F). In the classical picture of solid-state diffusion, anion spends the vast majority of the time rattling in its site, occasionally escaping andhopping to a neighboring vacant site through a polyhedral face, as illustrated on Fig.1a. The probability of this hopping event is described by transition state theory asactivated by thermal fluctuations of the host lattice, giving the Arrhenius equationfor temperature-dependent diffusivity:

D(T ) = D0 exp(−�Ea

/kT

)(1)

Activation energy �Ea corresponds to the height of the barrier in the energylandscape separating the two local minima sites. In the case of a single ion movingin a static periodic potential of an empty lattice, where the transport mechanismis known, this barrier can be computed readily using the nudged elastic band(NEB) method that uses a total energy model, such as density functional theory(DFT) to find the minimum energy path passing through the unstable saddle point(Jónsson et al. 1998). This approach, however, is only simple when Li ions aredilute interstitial defects in an ideal crystal lattice and do not interact while hoppingindependently of each other. In the best ionic conductors, Li-ion concentrationsare high, which means that Li ions are in close contact and Li-Li interactions arenot negligible. In such strongly interacting scenarios, forces confining Li ions in

4 B. Kozinsky

Fig. 1 (a) Ionic transport mechanism in Li garnets, showing Li migration from an oxygen-coordinated octahedral site to a neighboring tetrahedral site through a face. Geometric hull analysisdetermines that the occupancy changed and flags this as a hopping event (Courtesy of SnehaAkhade). (b) Voronoi decomposition of a crystal automatically identifies polyhedral sites that areused by the geometric transport analysis algorithm. The thickness of the graph edges connecting theVoronoi sites corresponds to the number of hops detected in the MD simulation analysis (Courtesyof Leonid Kahle)

their sites are counteracted by electrostatic repulsion from nearby Li ions, and Liions can displace each other. This leads to a self-blocking interaction where atmost one Li ion can occupy a polyhedral site. As a result, there is a significantamount of correlation between mobile ions, meaning that the energy landscapeof the Li sublattice (occupancy energies and transition barriers) is complex anddepends on local configuration. Configuration dependence of the energy barrier canbe introduced using the method of cluster expansions (Van der Ven et al. 2001),but also in this case, the transport mechanisms must be known a priori in orderto compute barriers in all configurations relevant for macroscopic transport. If it ispossible to obtain the energy barriers, the next step is to construct a kinetic MonteCarlo (KMC) model based on the complete catalog of all possible transitions andtheir probabilities.

An alternative practical approach to study complex ion conductors is moleculardynamics (MD), where interatomic forces are used to simulate the actual latticetime evolution according to Newton’s equations of motion, without taking asinput any information about transport mechanisms (Alder and Wainwright 1962).MD has emerged as a powerful tool for examining and optimizing solid ionicconductors, providing insights into transport mechanisms that are not possible toobtain experimentally. MD also relaxes the assumption of KMC models of time-independent transition probabilities, which allows realistic treatment of ion transportin dynamic environment such as liquids and polymers. At the same time, MDis significantly more computationally expensive and statistically inefficient thanstochastic methods, since transport is dominated by rare hopping events, and thermalvibrations consume the vast majority of computation time. Typical simulations usingab initio molecular dynamics, where forces are obtained from DFT, are limited toabout 200 atoms and 100 picoseconds taking roughly 2 weeks of wall time, using

Transport in Frustrated and Disordered Solid Electrolytes 5

the typical time step of 1–2 femtoseconds. The standard approach is to performsimulations at significantly elevated temperatures usually above 700 K, in order torapidly sample ionic motion, and then extrapolate diffusivity to room temperature,assuming that the Arrhenius dependence in Eq. (1) holds. This approach requirescaution when transport mechanisms are temperature dependent and if phase tran-sitions occur at lower temperatures. In classical MD, forces are determined fromanalytical force fields, such as Buckingham and Morse pair potentials, sometimeswith core-shell polarization and angle-dependent terms included. Classical MD hasthe advantage of being several orders of magnitude faster than ab initio dynamics.The limitation is that every material requires careful selection and validation of thefunctional form and parameters of the force field. Special care is needed especiallyfor ionic transport simulations, since diffusivity is determined by energies of low-symmetry configurations corresponding to transition states, while available forcefields are typically parameterized to represent equilibrium properties.

3 Transport Mechanism Analysis and Coarse-Graining

A key advantage of using molecular dynamics to study ionic transport in ceramicelectrolytes is the ability to directly characterize atomistic transport mechanisms,which consist of very fast and rare hopping events determining the long-rangemacroscopic transport character of the material. In order to extract informationabout the transport mechanisms, it is necessary to separate vibrational motion fromhopping events. The goal is therefore to map the dynamics of Li onto a discretelattice of sites in order to analyze the hopping sequences and statistics. One optionfor extracting hop events from the MD simulation is to keep a record for every Liof its nearest-neighbor atoms, based on a carefully chosen distance threshold, andto detect significant changes in the neighbor lists. This method is easy to implementbut has the disadvantage that trajectory discretization can depend on the distancethreshold and neighbor list, since different types of sites have different anioncoordination numbers. A more robust approach is to first determine the topology ofthe host lattice and partition it into nonoverlapping polyhedra formed by immobileanions. This can be obtained from crystallographic information and/or from Voronoidecomposition using anions as the vertices. Every Li ion is then assigned to anavailable polyhedral site in each MD snapshot. A simple way is to use a 3D convexhull construction (e.g., using the Qhull code) taking as input the positions of theLi ion of interest and all anion vertices of the candidate site. If the Li ion is insidea particular site, then the convex will contain only the anions, and if Li is outside,then its position will be part of the convex hull. This method is reliable, since thereare no free parameters, with the assumption that the host lattice does not rearrange.For example, in garnet ionic conductors, this analysis clarified that transport occursonly by tetrahedral to octahedral to tetrahedral site face-crossing hops, contraryto earlier conjectures of direct octahedral-octahedral edge-crossing hops (Fig. 1a).Using MD as a starting point for mechanistic investigations and coarse-grainingdynamics simulations is especially useful when transport mechanisms in a complex

6 B. Kozinsky

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Fig. 2 Result of geometric discretization of an ab initio molecular dynamics simulation of aLi-garnet structure performed at elevated temperature (800 K) in order to sample a significantnumber of hops. Endpoint configurations of detected transition paths are used as inputs into NEBcomputations of migration barriers

new material are not known a priori. In this way, the transport analysis can beautomated and used for high-throughput investigation of many diverse structuralvariations. In practice, this automatic procedure starts from high-temperature abinitio or classical MD to quickly sample important hopping scenarios. Once theMD trajectory is discretized, the site-to-site transition sequences and timings canbe extracted to obtain the complete transport mechanism sampled during the MDsimulation. Frequency statistics directly from MD are often insufficient to accuratelyestimate hopping probabilities, especially for the more rare events. In such cases,hopping rates for each event can be estimated using transition state theory, whichrelies on barrier height estimation using the NEB method. Finally, the catalog oftransition probabilities can then be used to define a coarse-grained dynamics modelsusing KMC, mentioned above, which simulates hopping as a stochastic Markovprocess while keeping track of real system time. However, this last step may not bestraightforward in some cases we consider below.

The resulting energy landscape determined by this analysis workflow containsimportant information about the nature of ionic transport. For example, Fig. 2shows an example of an MD trajectory of a garnet Li-ion conductor that was firstdiscretized using the geometric method described above and combined with NEBcalculation of barriers of the pathways connecting the extracted hop endpoints. It

Transport in Frustrated and Disordered Solid Electrolytes 7

is noteworthy that the resulting energy landscape is qualitatively different from theclassical picture of a single-ion hopping in a periodic potential landscape of the hostlattice. Instead, Li configurations explore a complex energy landscape in a collectiveway, and some events do not even exhibit barriers. Transport proceeds through asequence of moves involving multiple Li ions moving downhill or uphill in energy,and constructing a kinetic coarse-grained model may require an alternative approachto the conventional individual ion hop picture.

4 Correlation and Collective Motion

The outputs of both KMC and MD simulations are trajectories, or sequences ofatomic positions and corresponding energies and forces on each atom. Informationon positions (or velocities) can be used to calculate transport properties, hoppingmechanisms, and correlation functions. Ionic conductivity can be obtained usingthe Green-Kubo formalism, starting from

σ = N2q2

6kT Vlimt→∞

d

dt〈|R(t) − R(0)|〉 = q2

kT

N

VDσ (2)

where N is the number of diffusion Li ions, q = 1 is the charge, V is unit cellvolume, and R(t) is the vector connecting the center of mass of all Li ions withthat of the host material at time t. The reference to the host material center ofmass is especially important in simulations of multicomponent systems where thehost exhibits diffusive or fluctuating dynamics (Wheeler and Newman 2004). Thesecond equality in (2) is the Nernst-Einstein equation defining for conveniencea “conductivity” diffusion coefficient Dσ which, unlike conductivity itself, is notdirectly measured. A useful directly measurable quantity is the tracer diffusioncoefficient which is given by

D∗ = limt→∞1

6

d

dt

1

N

N∑i=1

⟨|ri (t) − ri (0)|2

⟩(3)

where ri are the positions of the Li ions and the quantity under the derivativeis called the mean square displacement (MSD). It can be shown that Dσ = D∗

when all cross-correlations between displacements of different Li ions vanish. Thisis a good commonly used approximation to obtain ionic conductivity from MDsimulations of dilute liquid electrolytes. However, in solid electrolytes, this is notthe case due to significant correlations, as mentioned earlier. The importance ofcorrelations can be estimated from the Haven ratio which is defined as H = D∗ /Dσ .Typical Haven ratios for fast Li-ion conductors range between 0.25 and 0.7. Theinverse of the Haven ratio has been interpreted as the average number of ionsthat interacts to move collectively (Doliwa and Heuer 2000). Despite the non-negligible effect of correlations, in practice MSD is the most frequently used

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Fig. 3 (a) String-like collective motion involving multiple Li ions in tetragonal Li-garnet oxides(Meier et al. 2014). (b) Effective flattened energy landscape (red) experienced by a string of ionsmoving in an external lattice potential (blue). Dilute ions or configurations commensurate with theexternal potential minima will experience much higher activation energies in their motion, makingcollective motions more likely

quantity for estimating ionic conductivity from MD since it converges much fasterwith simulation time than the fully correlated conductivity obtained from the centerof mass displacement. The reason is that in the first case, the estimate of the MSDis based on N-independent Li displacements, while in the second case, only thesingle displacement value of the center of mass is used to estimate the conductivity.Therefore, the statistical variance of the estimate of D* is smaller than that of theconductivity by a factor of order N2.

Haven ratio can be directly measured (e.g., from dc conductivity and NMRtracer diffusivity) but only provides a time-averaged indication of ionic correlations.To understand the microscopic nature of correlations, a detailed examination ofatomic trajectories is necessary, as discussed in the previous section. In some garnetcrystals, Li ions exhibit pronounced collective, or concerted, motion (see Fig. 3a),where multiple ions are displaced at the same time (Meier et al. 2014) rather thanindividual independent hops. In Li3PO4 this type of concerted motion was foundto be responsible for low activation energies of 0.2 eV, while direct hopping hasa migration barrier of 0.6 eV (Du and Holzwarth 2007). Concerted motion andhopping time correlations were also observed in LixLa2/3-x/3TiO3 (Catti 2011) aswell as sulfides and NASICON ion conductors using ab initio molecular dynamics(He et al. 2017). In each case, it was noted that collective migrations had lower-energy barriers compared to mechanisms involving single-ion hops.

To understand the appearance of collective modes, consider a string of ionsmoving in a single-particle potential energy surface with amplitude �Es, illustratedin Fig. 3b. If the ion-ion interaction is strong enough and constrains the ions to bespaced apart in a way that is not commensurate with the energy landscape maximaand minima, then the group as a whole experiences a smoother energy landscape,due to a “window-averaging” effect of the original energy landscape. Such collectivemodes, if they can form and move, will statistically dominate transport relative to

Transport in Frustrated and Disordered Solid Electrolytes 9

the individual ion hops that are much less frequent due to their higher barriers. Thissituation can only happen when nearest-neighbor Li interactions are significant, i.e.,when the fraction of occupied Li sites is high. This simple picture is an exampleof the phenomenon of frustration, as discussed in detail below, where strong short-range interactions prevent ions from settling into local minima, ensuring collectivemobility. It is interesting that collective “string-like” highly correlated motion hasalso been noted in other classes of materials, particularly in regimes close to meltingand glass transitions (Gebremichael et al. 2004). It is not clear whether collectivemotion is a necessary ingredient for high ionic mobility or whether it is simply thelowest-energy excitation in highly correlated situations. Their existence points tothe possibility that super-ionic conductors are close to a sublattice phase transitionwhere the Li sublattice exhibits liquid-like behavior, motivating the discussion ofphase transitions in the next section.

A practical way to quantify position and time correlations of ionic jumps and todetect such collective modes is the distinct part of the van Hove correlation function,defined as

Gd (r, t) = 1

N

∑i �=j

⟨δ(r − ∣∣ri (0) − rj (t)

∣∣)⟩ (4)

It describes the conditional probability of finding a pair of distinct atoms such thatthey are separated by distance of r after a time delay of t. Gd(r, t = 0) is justthe usual pair distribution function. Gd(r = 0, t) indicates probability of one ionbeing replaced by another ion in time t, which can be interpreted as a collectivesimultaneous event when t is comparable to the inter-site transition time scale. Inaddition to being useful for gaining intuition about ionic transport correlation, theVan Hove correlation function is measurable experimentally. Its Fourier transformis the intermediate scattering function that can be measured by neutron spin echospectroscopy. This quantity provides a valuable opportunity for direct validation ofMD simulations against experimental data.

5 Geometric Sublattice Frustration

The overall ability of the Li sublattice configuration to rearrange with low transitionbarriers is the necessary ingredient of high ionic conductivity. Since ionic transportis a statistical thermally activated phenomenon, the probability of such rearrange-ments is proportional to the number of available configurations. It is clear, therefore,that in order to maximize conductivity, we need to maximize the sublattice entropy.As with any system in thermal equilibrium at temperature T, the ionic conductorwill minimize the free energy and therefore tend to find low-energy configurationsof the Li sublattice. If the ground state is well ordered, the only available mechanismfor transport is through creation of mobile defects. In regular periodic crystals, theactivation energy required for ionic hopping Ea = Edef + Ehop is a sum of the energy

10 B. Kozinsky

required to first create a mobile defect and the migration barrier height. However,thermal energy scale at room temperature is rather low, with kT only on the orderof 25 meV, making it difficult to overcome strong ionic interactions and disturb astable ground state configuration in a crystal. This is the reason that in crystallineionic solids, diffusion is much slower than in liquids. The goal in designing afast ionic conductor, therefore, is to ensure that the ground state is intrinsicallydegenerate or nearly degenerate, so that it is able to rearrange with minimal energycost. In materials science terms, this qualitatively corresponds to engineering a highintrinsic concentration of mobile carriers by varying the composition and structure.In mathematical terms, this can be described as symmetry breaking and maximizingthe amount of geometric frustration in the Li sublattice, preventing the system fromsettling into a low-entropy arrangement.

We generally define frustration as competition between local interactions(Li-Li and Li-anion), resulting in near degeneracy and disorder in the groundstate configuration of the Li sublattice. Locally, this situation comes about when Lioccupancy of a lattice site cannot be decided based on minimizing total interactionenergy with its neighbor sites. Globally, frustration manifests itself when there isnot a single way to minimize energy by ordering mobile ions on their sublattice.

Intuitively, the role of frustration is to flatten the energy landscape of Li ions,preventing locking of mobile carriers in deep energy minima corresponding to stableordered arrangements (Fig. 3b). Li ions in the frustrated sublattice are then forced toexplore the large number of degenerate states in a narrow energy window. Effectivefrustration, which maximizes the sublattice entropy, results in a high concentrationof charge carriers and high interstate transition rates, the two necessary ingredientsfor fast ion conduction.

Geometrically frustrated systems have a long history in solid-state physics, tradi-tionally studied in the context of antiferromagnetic spin models on various latticeswith short-range interactions. Due to the rich physics and emergence of unexpectedphenomena, frustration in magnetism is an increasingly active research field incondensed matter theory (Balents 2010). Complexities arising from the presenceof extensive ground state degeneracy lead to unusual thermodynamic and transporteffects. These include suppression of finite temperature order-disorder freezingtransitions, giving rise to disordered spin liquid ground states. Recently attention tofrustrated lattice systems has increased due to the emergence of Majorana fermions,topological quantum excitations resembling relativistic particles (Kitaev 2006).

In order to illustrate the concept of frustration and its implication on chargetransport, let us consider a simple model: a two-dimensional antiferromagneticIsing model with nearest-neighbor interactions on a triangular lattice. As illustratedon Fig. 4a, once opposite spins occupy two of the three corners of a triangle,there is no unique choice for the third spin direction that minimizes the energy.To connect this peculiar behavior to charge transport, we can map the lattice spinvariables to charge occupancy, so that spin up corresponds to occupied site and spindown to unoccupied, as shown on Fig. 4a. The antiferromagnetic spin interactionphysically corresponds to electrostatic repulsion of charges on neighboring sites.This construction is representative of a class of lattice gas models with on-

Transport in Frustrated and Disordered Solid Electrolytes 11

Fig. 4 (a) Frustrated configuration of spins on a triangle is mapped onto site occupancy in a latticegas model. The two configurations have the same energy if only nearest-neighbor interactions areconsidered. (b) One of many possible ground state configurations at ½ occupancy of the triangularantiferromagnetic Ising model. (c) A well-ordered low-entropy ground state configuration existsfor occupancy fractions 1/3. At slightly higher occupancy, entropy increases, since additionalcharges can freely move through the ordered background, forming a dilute gas

site exclusive constraints. Frustrated lattice models are commonly discussed inthe context of spin or electron charge degrees of freedom, but it is natural toextend this understanding to crystalline ionic conductors. In fact, the triangularlattice gas model describes Li-ion configurations in transition metal layered oxidesLix(Ni,Co,Mn)O2 (0 < x < 1), which are by far the most commonly used cathodematerials in Li-ion batteries today. In these materials, the crystal structure is made upof layers of oxygen-coordinated edge-sharing octahedral sites filled with transitionmetals, separated by layers of octahedral Li sites that form a triangular lattice, suchthat Li ions move in and out of the layers during battery cycling.

Relevant temperature scales in this lattice gas model are approximately given interms of the nearest-neighbor interaction strength V. The simplest to understand isthe high-temperature behavior kT >> V, corresponding to a disordered gas phasein which charges can move via independent hops, only subject to the exclusivesite occupancy constraint. At low temperature kT << V, the system settles into theground state configuration (solid phase). At intermediate temperatures, the system

12 B. Kozinsky

leaves the ground state manifold, and a population of mobile defects exist, e.g., allthree triangle corners occupied or empty.

Due to local frustration, the ground state at half-filling (n = 1/2, i.e., zeroaverage magnetization) is highly degenerate, allowing many configurations withthe same lowest energy (illustrated in Fig. 4b). This model with only nearest-neighbor interactions can be solved exactly to show that there is actually no orderingphase transition at any finite temperature (Wannier 1950). In other words, the spinlattice never truly freezes and remains in a glassy state. We should note that thisis true because the nearest-neighbor antiferromagnetic interaction is incompatiblewith the triangular lattice symmetry in the case of n = 1/2, but there exists a well-ordered ground state at n = 1/3, where a pronounced freezing transition occurs atfinite temperature and consequently entropy and transport are strongly suppressed.This emphasizes that “local frustration” does not necessarily guarantee groundstate degeneracy. Instead, ground state degeneracy is a function of both carrierconcentration and global lattice symmetry, as elaborated further in the next section.

Remarkably, the frustrated ground state without any “defect” excitations has sub-stantial ability to rearrange, due to local charge motions. Monte Carlo simulationsof thermodynamics and transport indicate that this system appears to possess asufficient amount of residual entropy in the n = 1/2 ground state to allow for finiteconductivity, even in the presence of long-range interactions (Levitov and Kozinsky1999). When long-range screened Coulomb interactions are included for n = 1/2,the degeneracy is broken, and an ordering phase transition appears but at muchlower temperature than in the case of n = 1/3, indicating that frustration effectivelysuppresses freezing. In this regime, charge transport has collective character ofa correlated fluid due to strong short-range correlations between charges whichconstrain individual charge migrations, preserving the degenerate ground statemanifold of configurations. The change in conductivity behavior while coolingdown from the disordered into the correlated fluid and solid phases can also beseen in Fig. 5. The low-temperature curves indicate that the conductivity vanishesmore quickly near the densities of a simple fraction form (n = 1/4, 1/3, 1/2). Thiscorresponds to freezing of the system into states commensurate with the lattice atthese values of n. The qualitative features of the phase diagram and its influenceon charge transport do not seem to be affected by including disorder (Novikov et al.2005). So far the model only considers energy of configurations and does not includekinetic details of inter-site transition. Even when more detailed hopping mechanismsare included specifically to describe transport of Li in LixCoO2-layered oxidesusing cluster expansions and kinetic Monte Carlo, the main effects of concentrationdependence of conductivity are preserved (Van der Ven et al. 2001). This simple2D model demonstrates the key principle that charge transport without defectexcitations is possible in an ideal symmetric lattice, as long as frustration exists thatestablishes degeneracy and disorder at non-commensurate carrier concentrations. Ina later section, we will see that this principle allows us to understand and designmore complex ionic conductors.

Transport in Frustrated and Disordered Solid Electrolytes 13

Fig. 5 The scaled conductivity in the triangular lattice gas model with long-range screenedCoulomb interaction, as a function of charge occupancy fraction n for several temperatures. Thetemperature values are given in the units of V(a), the nearest-neighbor interaction. Arrows markthe features corresponding to the freezing phase transitions at n = 1/4, 1/3, 1/2. Dashed linecorresponds to the high-temperature limit. The symmetric high-occupancy half of the concentrationrange is not shown (Levitov and Kozinsky 1999)

5.1 Algebraic View of Sublattice Frustration

We now turn to the discussion of ways to identify and design materials withfrustrated mobile sublattices. In mathematical terms, the appearance of a strongdegeneracy of the ground state requires that there is no low-entropy highlysymmetric configuration commensurate with the host lattice. At the same time,frustration arises due to strong short-range interaction, which requires highLi-ion concentration in the material. From this point of view, when optimizingionic conductivity of a particular crystal, Li concentration is the key tuningparameter. In the light of the discussion in the previous section, our goal is toidentify high-symmetry ground states of the sublattice configuration, into which Liions can freeze. Ability to do this has wide applicability in general, especially forLi-containing crystals, since Li has a very weak X-ray scattering cross-sectionand Li site occupancies are very difficult to assign without resorting to neutrondiffraction of single-crystal samples. The nature of ionic conduction is also verydifficult to observe, only indirectly using solid-state NMR techniques. In thecontext of fast ion conductors, knowledge of where ordered configurations appear

14 B. Kozinsky

in the composition space is critical to guide optimization away from these Liconcentrations, thus maximizing frustrated disorder.

In order to systematically and rigorously identify ordered ground state configura-tions as a function of Li occupancy, group theoretic tools are particularly effective.We illustrate this approach in the case of Li-garnet family of ionic conductingcrystals with room temperature ionic conductivity of up to 1.6 mS/cm. Combinedwith its electrochemical stability in contact with Li metal and high mechanicaltoughness, this makes it an attractive and widely investigated candidate for a solidelectrolyte in a Li metal battery. The prototypical garnet structure is body-centeredcubic (group Ia-3d No. 230) with the chemical formula LixB3C2O12. This structureis remarkably robust in regard to changes in the B and C cation composition(B = La, Ca, Ba, Sr, Y, Pr, Nd, Sm-Lu, and C = Zr, Ta, Nb, Nd, Te, W) and isable to accommodate Li concentrations x ranging from 3 to 7 per formula unit. Inthe vast majority of crystalline oxides, Li ions occupy octahedral and/or tetrahedralsites. Garnets are no exception; there are two distinct sets of sites: octahedral(O, Wyckoff position 48 g) and tetrahedral (T, at position 24d). The numbersindicate multiplicities, i.e., 48 O sites are equivalent and can thus be either all filledor all empty in the prototype group of the host lattice. Each T site shares a facewith four O sites, while each O site shares a face with two T sites, thus forming apercolating Li site network of pathways for ionic conduction.

The arrangement of Li ions among available crystallographic sites is the maindegree of freedom that controls ionic transport. However, complete brute-forceanalysis of Li orderings is impossible: for instance, in the simple cubic cellrepresentation of the x = 7 garnet, there are 72 Li sites, of which 56 are occupied,resulting in 1015 possible arrangements. In order to deal with the combinatorialspace of configurations, it is useful to apply crystal group analysis to classifysimply ordered Li arrangements for each composition x. The procedure starts withexhaustive crystallographic projections of the prototype high-symmetry group of thehost lattice (Ia-3d No. 230 in case of garnets) onto lower-symmetry subgroups. Thisallows to split the high-symmetry Wyckoff positions (48 g and 24 d), correspondingto all filled or empty T and O sites, into sets of smaller multiplicities, so thatthey can be separately occupied. The resulting structures are the highest symmetryconfigurations, commensurate with the host lattice space group, where Li occupiesa fraction of the sites. For instance, group No. 198 P213 is a cubic subgroup of theparent group, where the T site with the original prototype Wyckoff position 24dis split into two Wyckoff positions 12b corresponding to two subsets of 12 sitesrelated by symmetry operations. Each subset can then be occupied independently.There are many ways of breaking the parent prototype symmetry, and going downto the triclinic group P1 corresponds to removing all symmetry and enumerating allpossible configurations, which is not feasible. Therefore, it is practical to constrainthe search to structures that have high enough symmetry to be likely groundstates and can be represented using small unit cells in an ab initio simulation.Mathematically this corresponds to calculating the tree of maximal subgroups of theparent prototype group of the host lattice, limiting them to certain Bravais lattices(e.g., only cubic or tetragonal) and a low k-index. The k-index of a subgroup is

Transport in Frustrated and Disordered Solid Electrolytes 15

Fig. 6 (a) Illustration of symmetry breaking preserving square symmetry in a larger supercell (k-index 4). Different colors represent sets of equivalent positions that can be removed or displacedaccording to the subgroup Wyckoff positions. (b) Partial group-subgroup relationship tree thatexemplifies a typical set of paths from the parent group No 230 to lower-symmetry subgroups.T and O represent tetrahedral and octahedral sets of sites, respectively, together with theirmultiplicities

the multiplication factor relating the volume of the primitive cell of the subgroupwith respect to the primitive cell of the original prototype structure. We can thenanalyze the splittings of Wyckoff positions in each subgroup and conjugacy classin order to identify symmetrically related sets of Li sites and their multiplicities.A simple 2D example of crystal symmetry breaking is illustrated on Fig. 6a. Thisprocedure has the advantage of rigorously evaluating all possibilities starting withthe highest symmetry and usually results in only a few hundred high-symmetryconfigurations with various Li concentrations, a significant reduction compared tothe original space of possibilities.

To implement the subgroup projection in practice, it is convenient to use symme-try tools provided by the Bilbao Crystallographic Server (Aroyo 2006). For the caseof garnets, a partial tree of maximal subgroup projections down to orthorhombicsymmetry and k-index 2 is illustrated on Fig. 6b, which lists symmetry equivalentsets of sites with their Wyckoff multiplicities. Crystal structures are constructed ofthe simple cubic cell where each independent set of Wyckoff positions is eitherincluded or excluded in the structure. The total energy of each configuration iscomputed using density functional theory. For the garnet family, the result is thatthere are unique low-energy configurations at certain integer values of Li compo-sition (x = 3, 4, 6, 7). High-symmetry configurations also exist at other integercompositions but are very close in energy (less than 3 meV per atom), resulting indisordered structures at room temperature. At non-integer compositions, no low-

16 B. Kozinsky

1.0E-09

1.0E-07

1.0E-05

1.0E-03

1.0E-01

1.0E+01

3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00

Ioni

c co

nduc

tivity

(S c

m-1

)

Li concentration (x)

500 K 1700 K

c-Li7

t-Li7

EA (1700 - 1100 K) = 0.273 eV

EA (900 - 500 K) = 0.983 eV

-30

-25

-20

-15

-10

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Log

(D*)

1/T × 10 3 (1/K)

a)

b)

-

Fig. 7 (a) Ionic conductivity as a function of Li content x in the garnet crystal, from MDsimulations. Red triangles indicate compositions at which ordered ground states were identifiedby the subgroup projection method. (b) Logarithm of the tracer diffusion coefficient versus inversetemperature for the x = 6 structure, illustrating the order-disorder phase transition at about 900 K.Black circles represent high-temperature values, and green triangles represent low-temperaturevalues

entropy ground states are found at all by the symmetry analysis, at least within theconstraint of a cubic structure with fewer than 200 atoms in the unit cell. SinceLi-Li interactions are weak beyond the length scale of the already large unit cell(13

′Å), this constraint is physically justified. Compositions where ground states

have high degeneracy and frustration due to symmetry incompatibility are expectedto have the highest ionic conductivity. Indeed, as illustrated in Fig. 7a, conductivityis noticeably suppressed near precisely the compositions where well-ordered groundstate Li configurations exist, with non-negligible excitation energies. On the otherhand, much shallower energy landscapes and degeneracy result in better ionic

Transport in Frustrated and Disordered Solid Electrolytes 17

mobility in disordered configurations that arise in situations of frustration, whereorder-disorder phase transitions are suppressed (Kozinsky et al. 2016), and muchlower conductivity when order-disorder transitions appear at high temperature, e.g.,at x = 6 (Fig. 7b).

6 Extrinsic Disorder: Amorphous Materials and RandomAlloys

An alternative way to avoid ordered Li sublattice configurations is to break thetranslational symmetry of the host lattice. Extrinsic disorder can be implementedin several ways. By melting and rapidly cooling, some crystalline materials can beturned into glasses that remain in the state of quenched disorder indefinitely at roomtemperature. This process typically preserves the local coordination and bondingfeatures of the crystal but destroys the long-range order. Examples of materialswhere ionic conductivity in crystalline form is significantly lower than in amorphousstructure are Li3PO4 and LiAlSiO4. Geometric frustration between the buildingblocks of the host lattice (not Li sublattice as discussed earlier) can result in aglassy disordered morphology of the material (e.g., water ice). Disorder can also beintroduced by random site occupancy in regular crystals, preserving the long-rangelattice structure. For instance, experiments and MD simulations show that randomalloying in Ba1−xCaxF2 enhances F-ion conductivity by a factor of 105 (Duvel etal. 2017). Similarly, classical molecular dynamics indicated that Li4SiO4-Li3PO4crystalline alloy mixtures exhibit 2–4 orders of magnitude higher ionic conductivitythan either endpoint (Deng 2015).

The effect of extrinsic host disorder on ionic transport is not well understood,despite significant interest (Dyre et al. 2009), and numerical modeling plays acentral role in this direction. Open questions remain about the nature of order-disorder phase transitions in lattices with quenched extrinsic disorder. A usefulmodel to represent extrinsic quenched disorder is the “random-field Ising model,”where each site experiences a random local chemical potential (Berretti 1985).For strong enough disorder, suppression and eventual elimination of order-disordertransitions are observed, but the cause is very different from geometric frustration.Monte Carlo simulations are used to examine different regimes that are not coveredby analytical renormalization group techniques, far away from critical points.There is also incomplete understanding of how quenched disorder relates to ionictransport. Qualitative features of transport in disordered systems are illuminatedusing the random barrier model (Dyre 1988). In this model, a single particleis hopping on a lattice, where barrier heights between sites are drawn from asmooth probability distribution. In 2D and 3D, long-range transport happens alongpercolation paths composed of hops with low migration barriers. Thus diffusionat low temperatures is controlled by a single critical percolation energy barrier,explaining why Arrhenius behavior is observed in systems with a wide range ofactivation energies. In large-scale MD simulations of disordered systems, care mustbe taken to ensure that tracer diffusivity is sampled for long enough time for the

18 B. Kozinsky

system to equilibrate and reach a truly long-range diffusive transport regime, asopposed to Li exploring a confined percolation cluster in a sub-diffusive mode.This can be done by checking whether the mean square displacement increaseslinearly with time, as expected from Fick’s law and Einstein relation. At highertemperatures, a wider range of pathways and a larger fraction of activated mobileions may start contributing to the transport. In experiment deviations from Arrheniusbehavior are observed in glass ionic conductivity at high temperatures, but thereason is still not well understood (Kincs and Martin 1996).

6.1 Host Dynamics

In addition to static extrinsic disorder, in some of the best ion conductors, the hostitself has mobile degrees of freedom that couple in nontrivial ways with the mobileions. Host dynamics on the time scale of Li migration can effectively introducefluctuations of the Li configuration energy landscape, preventing long-term lockingin local minima. This effect is well known in liquid electrolytes, where solvationshells of mobile species are dynamic, resulting in very high diffusivity values. Themost widely studied solid electrolytes relying on host dynamics are polymer saltsolutions, particularly the family of polyethylene oxide (PEO)-based polymers. Inthese electrolytes, Li ions are coordinated by the polymer, and conduction mainlyhappens due to segmental motion of the flexible chains, with Li occasionallyhopping from one chain to another. As a result, PEO polymer electrolytes onlyhave substantial ionic conductivity above their glass transition temperature, whenthe host polymer has significant local mobility, similar to a viscous liquid. Dueto the dynamic nature of the host, molecular dynamics is the only option forsimulation, and due to the size of these amorphous systems (>5000 atoms) and longequilibration times (>10 ns), classical molecular dynamics is the method of choice,typically using force fields developed for organic molecules. In some inorganicmaterials, dynamic host behavior is also important in assisting ionic transport bypreventing mobile ions from ordering and by increasing the rate of exploration ofthe configuration landscape. In oxides host dynamics are rare due to strong ionicinteractions, but some examples are found in covalently bonded oxide polyanionmaterials. These types of dynamic mechanisms are material-specific, and due to themixed ionic-covalent nature of bonding, ab initio molecular dynamics is mostly usedto study the interplay between structure dynamics and ionic transport. For instance,high proton conductivity is demonstrated by ab initio MD to be mediated by fastSO4 anion in-place rotations in CsHSO4 proton electrolyte (Wood and Marzari2007). In the recently discovered class of closo-borate fast ion conductors suchas Li2B12H12, the (BH)x

2− anion centers are ordered on a regular lattice due tolong-range electrostatic interaction, but their icosahedral geometry is incompatiblewith the lattice symmetry (Kweon et al. 2017). This geometric frustration causesthe anions to exhibit orientation disorder and in-place rotation. Li ions jump in theextremely dynamic energy landscape, driven by its fluctuations, and explore a large

Transport in Frustrated and Disordered Solid Electrolytes 19

number of available dynamic sites. This type of “dynamic frustration” is a promisingmechanism to enhance ionic mobility explore and requires further understanding.

7 Outlook

There is converging understanding from recent molecular dynamics and experi-mental work that ionic transport in fast Li-ion conductors at room temperatureis different from the classical picture of independent migrating defects and thatconcepts based on single particles are not entirely relevant. Evidence of importanceof many-particle collective effects comes from molecular dynamics simulationsand from measurements of the Haven ratio. Correlated transport phenomena areprominent, but their origins need deeper microscopic and statistical understanding.In a surprising number of cases, mobile sublattice disorder seems to dramaticallyimprove ionic conductivity, whether it is due to frustration, extrinsic distortion andalloying, or dynamic coupling to the host fluctuations. In particular, Li sublatticefrustration with maximal residual entropy may be a promising recipe for high ionicconductivity at low temperature, due to both high carrier concentration and theircollectively mobile behavior. The reasons and mechanisms are not yet well knownand may not be the same in different materials, but the complex regimes of highLi-ion concentration and strong interaction deserve a general systematic con-sideration. This further elevates the importance of explicit molecular dynamicsapproaches and development of fast transferable ways to accurately simulatecomplex ion dynamics, especially dealing with rare hopping events. Toward the goalof establishing a common framework of understanding room temperature fast iontransport, we underscore the relevance of rigorous methods developed in other fieldsfor describing sublattice order-disorder phase transitions and transport in correlatedliquids and disordered media.

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