# The effect of cation-disorder on lithium transport in halide superionic
conductors

Peichen Zhong Department of Materials Science and Engineering, University of
California, Berkeley, California 94720, United States  Materials Sciences
Division, Lawrence Berkeley National Laboratory, California 94720, United
States  Sunny Gupta Department of Materials Science and Engineering,
University of California, Berkeley, California 94720, United States  Materials
Sciences Division, Lawrence Berkeley National Laboratory, California 94720,
United States  Bowen Deng Department of Materials Science and Engineering,
University of California, Berkeley, California 94720, United States  Materials
Sciences Division, Lawrence Berkeley National Laboratory, California 94720,
United States  KyuJung Jun Department of Materials Science and Engineering,
University of California, Berkeley, California 94720, United States  Materials
Sciences Division, Lawrence Berkeley National Laboratory, California 94720,
United States  Gerbrand Ceder<EMAIL_ADDRESS>Department of Materials
Science and Engineering, University of California, Berkeley, California 94720,
United States  Materials Sciences Division, Lawrence Berkeley National
Laboratory, California 94720, United States

###### Abstract

Among the chloride-based Li-ion solid electrolytes, Li2ZrCl6 (LZC) have
emerged as potential candidates due to their affordability, moisture
stability, and high ionic conductivity. LZC synthesized by solid-state heating
exhibits limited Li-ion conductivity while the mechanochemical ball-milled
material is more conductive. In this computational study, we integrate
thermodynamic modeling, using cluster-expansion Monte Carlo, and kinetic
modeling, using molecular dynamics, to investigate whether cation disorder can
be achieved in LZC, and how it affects Li-ion transport. Our results indicate
that fast Li-ion conductivity is induced by the activation of Li/vacancy
disorder, which itself depends on the degree of Zr disorder. We find that the
very high-temperature scale at which equilibrium Zr-disorder can form
precludes any equilibrium synthesis processes for achieving fast Li-ion
conductivity, rationalizing why only non-equilibrium synthesis methods, such
as ball milling leads to good conductivity. We identify as the critical
mechanism the lack of Li/vacancy disorder near room temperature when Zr is
well-ordered. Our simulations further show that the Li/vacancy order-disorder
transition temperature is lowered by Zr disorder, which is necessary for
creating high Li diffusivity at room temperature. The insights obtained from
this study raise a challenge for the large-scale production of these materials
and the potential for the long-term stability of their properties.

###### pacs:

##  I Introduction

Figure 1: (a) A summary of the synthesis paths and ionic conductivities of
LZCs. (b) An illustration of the ground-state structure and primitive cell
used for lattice models (b) $\alpha$-LZC (hcp) and (c) $\beta$-LZC (fcc). The
green/orange/white balls represent the cation site that can be occupied by
Li/Zr/vacancies, respectively. The grey ball represents the anion site with Cl
occupancy.

High-Li-ion-conductivity solid electrolytes are essential for realizing all-
solid-state batteries (ASSBs), which offer the potential for increased
volumetric energy density and improved safety compared to conventional liquid-
electrolyte-based batteries [1, 2, 3]. Chloride-based Li-ion solid
electrolytes, with the general formula LixMyCl6, where M represents a
transition metal and in which Cl anions form a close-packed framework, have
become of interest as catholytes in ASSBs [4, 5, 6, 7, 8, 9, 10]. Many of
these electrolytes exhibit high room-temperature (RT) Li-ion conductivity
($>1$ mS/cm), good interfacial stability with cathode materials, and high
oxidation stability [11, 12, 13, 5, 9], consistent with early theory
predictions [14]. Significant challenges still exist before these materials
can be commercialized in practical cells. Some of the halide conductors
contain low-abundance metals (e.g., Sc/Y/Tb/Lu). In addition, many of the
interesting halides are unstable in ambient conditions and their ionic
conductivity significantly degrades by exposure to moisture [15, 4]. Recently,
the Zr-based compounds (e.g., Li2ZrCl6, LZC) and their doped variants have
demonstrated the potential to overcome these challenges [16, 17, 18, 19, 7,
20, 21, 22]. LZC exhibits distinct polymorphism with its structure depending
on the synthesis method as summarized in Fig. 1(a). Mechanochemical synthesis
produces $\alpha_{h}$-LZC (hexagonal close-packed anions, hcp), whereas the
$\beta_{c}$-LZC (cubic close-packed anions, ccp) can form by annealing
$\alpha_{h}$-LZC at temperatures above 260 ∘C (see Fig. 1(a)). While the ball-
milled $\alpha_{h}$-LZC exhibits RT Li-ion conductivity of approximately 0.8
mS/cm [16], the annealed LZC in either the $\alpha$\- or $\beta$-phase display
significantly lower RT Li-ion conductivity [16]. This has led to the
hypothesis that some disorder is critical to achieve high ionic conductivity.
Mechanochemical ball-milling as a non-equilibrium synthesis method is capable
of inducing high-energy cation-disordered configurations [23, 24]. Schlem  _et
al._ [25] demonstrated that in Li3YCl6 and Li3ErCl6 mechanochemical synthesis
induces cation defects by creating disorder on the sites occupied by Y/Er and
that this results in a reduced activation energy and enhanced ionic
conductivity. Using molecular dynamics simulations, Wang  _et al._ [26]
revealed that the Li sublattice disorders above $T_{c}=425$ K in Li3YCl6, and
that the associated broadening of the Li site energies creates fast Li-ion
diffusion. Other approaches such as introducing stacking faults [27] and
tunning Li/metal stoichiometry to create percolation pathways [28, 29] have
also been proven to enhance Li-ion conductivity.

In this study, we seek to explain the correlation between synthesis route,
cation disorder, and ion transport in LZC systems through ab initio modeling.
Using cluster-expansion Monte Carlo (CE-MC) simulations for both $\alpha$\-
and $\beta$\- phases, we demonstrate that, in equilibrium, Li- and Zr-disorder
occur at very different temperatures. Our results suggest that facile Li
disorder only emerges at room temperature in the presence of Zr disorder, but
that equilibrium Zr disorder only occurs at very high temperatures. In the
low-temperature equilibrium states in which Zr is well-ordered, Li/vacancy
intersite exchanges are thermodynamically limited, leading to very low
conductivity. The finding that at low temperature Li disorder is driven by Zr
disorder is further confirmed by molecular dynamics (MD) simulations using
machine-learning interatomic potentials, in which we find a significant
improvement in Li ionic diffusivity with the presence of Zr disorder. Our
investigation demonstrates that Zr disorder is essential to achieve high Li
conductivity, but cannot likely be achieved through thermal equilibrium
synthesis routes, thereby explaining why extensive ball milling is required
for these materials. We believe this to be a challenge for the large-scale
production of these materials as well as a potential problem for the long-term
stability of their properties.

##  II Methods

###  II.1 Cluster expansion

The cluster-expansion (CE) method [30] is used to study the configurational
thermodynamics of materials in which sites can be occupied by multiple
cations, and has been applied to study the Li-vacancy intercalation chemistry
in layered and disordered materials [31, 32] as well as solid electrolytes
with Li/vacancy disorder [33, 34]. The CE expands the energy as a sum of many-
body configurational interactions:

$E(\boldsymbol{\boldsymbol{\sigma}})=\sum_{\beta}m_{\beta}J_{\beta}\left\langle\Phi_{\boldsymbol{\alpha}\in\beta}\right\rangle_{\beta}+\frac{E_{0}}{\varepsilon_{r}},~{}\Phi_{\boldsymbol{\alpha}}=\prod_{i=1}^{N}\phi_{\alpha_{i}}(\sigma_{i})$
(1)

A configuration $\boldsymbol{\sigma}$ represents a specific occupancy state of
species its allowed sites, and $\sigma_{i}$ describes which species sit on the
$i$-th site of the lattice. The cluster basis function
$\Phi_{\boldsymbol{\alpha}}=\prod_{i=1}^{N}\phi_{\alpha_{i}}(\sigma_{i})$ is
the product of site basis functions $\phi_{\alpha_{i}}(\sigma_{i})$ across a
collection $\alpha$ of multiple sites. The average is taken over the crystal
symmetry orbits $\beta$, forming a complete basis to expand the energy
function in the configuration space. The expansion coefficients $J_{\beta}$
are called effective cluster interactions (ECIs). The electrostatic energy
(Ewald energy $E_{0}/\varepsilon_{r}$) is included to capture long-range
electrostatic interactions ($E_{0}$ is the unscreened electrostatic energy,
and $1/\varepsilon_{r}$ is fitted as one of the ECIs).

The CEs of LZC were performed with pair clusters up to 8 Å, triplet clusters
up to 5 Å, and quadruplets up to 5 Å using a hcp primitive cell based on the
unit cell of Li3YCl6 (ICSD-29962) for $\alpha$-LZC, and with pairs up to 10 Å,
triplets up to 7 Å, and quadruplet interactions up to 4 Å using a fcc
primitive cell based on the unit cell of Li3InCl6 (ICSD-17638) for $\beta$-LZC
(see Fig. 1). Occupancy of the metal sites is restricted to Li+/Zr4+/vacancy
in the CEs. In total, 104/70 ECIs (including the constant term $J_{0}$) were
defined, and the CEs for the MC simulations were fitted with the energies of
528/568 different structures for $\alpha$/$\beta$-LZC, where 62/82 structures
were directly relaxed using r2SCAN density functional theory (DFT)
calculations, and the energy of the remaining structures was determined by
CHGNet through the ground-state determination procedure described below. When
CHGNet is tested on the DFT relaxed structures, a root-mean-squared error of 8
meV/Cl is achieved. Although the CHGNet energy is not as accurate as the
direct DFT value, we find that the CHGNet error is similar to the CE fitting
error ($<7$ meV/Cl). It is noted that this is a typical error scale for
describing the configurational disorder as reported in several reported solid-
state systems [35]. An $\ell_{0}\ell_{2}$-norm regularized regression with
ground-state preservation algorithm was applied when fitting the ECI to
address the complexity-induced over-fitting [36, 37]. Gurobi was used as the
optimizer [38]. We refer the readers to Ref. [39] for detailed formalism and
terminology of the CE. The ECIs were parameterized with root-mean-squared
errors $<7$ meV/Cl for both the $\alpha$\- and $\beta$-CEs. The CE
constructions were performed using smol [40] and pymatgen for structure
processing [41].

###  II.2 Ground-state determination

The ground-state (GS) determination identifies the metal configuration with
the lowest energy among all possible configurations on the partially occupied
lattice. To efficiently accomplish this task, we relied on the Crystal Graph
Hamiltonian Neural Network (CHGNet) to relax structures and evaluate their
energy [42]. The CHGNet was fine-tuned on a dataset in the Li–Zr–Cl chemical
space, which is composed of DFT structure relaxation trajectories of LZCs,
LiCl, and ZrCl4 (approximately 19,000 atomic configurations). The different
Li/Zr/vacancy orderings in LZCs were generated using the enumeration method
(OrderDisorderedTransformation) in pymatgen [41]. CHGNet achieves mean
absolute errors of 6 meV/atom for energy, 11 meV/Å for interatomic forces, and
0.011 GPa for stress on the validation set. To find the ground state of LZC,
we implemented a GS search workflow, which is composed of the following steps.
(1) Fitting a CE Hamiltonian using the CHGNet-relaxed structures as a training
set, which is obtained by collecting the relaxed structures from step (3)
cumulatively as the iteration proceeds (random values of ECIs were initialized
in the first iteration); (2) running simulated annealing with canonical MC to
search for any structure with lower energy compared to the existing CE
training set; (3) Each such structure was relaxed using CHGNet and added to
the set of structures to which the CE is fitted, after which all steps were
repeated from (1). The workflow was fully automated until no new GS was
detected. After that, the 50 structures with the lowest energy from the CHGNet
predictions were relaxed with DFT. The ground state ordering was determined to
be the structure with the lowest DFT energy.

DFT calculations were performed using the regularized strongly constrained and
appropriately normed meta-GGA exchange-correlation functional (r2SCAN) [43,
44], as implemented in the Vienna ab initio simulation package (VASP) using
the projector-augmented wave method [45, 46], a plane-wave basis set with an
energy cutoff of 680 eV and a reciprocal space discretization of 25 k-points
per Å-1. The calculations were converged to $10^{-6}$ eV in total energy for
electronic loops and 0.02 eV/Å in interatomic forces for ionic loops.

###  II.3 Monte Carlo simulations

Equilibrium metal disorder at finite temperatures can be studied by MC
simulations on the CE Hamiltonian. These MC simulations were performed in the
canonical ensemble by heating from the identified GS (see Fig. 1(b)-(c)) at
$T=0$ K to 5000 K with a temperature step of $\Delta T=50$ K. At each
temperature step, a total of 200 structures were uniformly sampled from
2,000,000 configurations obtained in the canonical equilibrium ensemble after
equilibration at each $T_{\text{eff}}$ with 200,000 MC steps. The structures
sampled from the equilibrium ensemble were used to calculate the average
energy $\langle E\rangle$ and the heat capacity
$C_{v}=\sigma^{2}(E)/k_{B}T^{2}$, $\sigma^{2}(E)=\langle E^{2}\rangle-\langle
E\rangle^{2}$ is the energy fluctuation. Randomly picked MC structures from
the equilibrium ensembles at different $T_{\text{eff}}$ were selected to study
the equilibrium Li/vacancy disorder for fixed Zr occupancies. In these
simulations, the Zr disorder from the MC state obtained at $T_{\text{eff}}$ is
frozen but the Li/vacancy configuration is equilibrated from $T_{\text{eff}}$
to 0 K with a temperature step of $\Delta T=-50$ K.

Figure 2: (a) The average potential energy above the ground state ($\Delta
E=E(T_{\text{eff}})-E_{\text{GS}}$) sampled full equilibrium CE-MC
simulations. The green/orange dots represent $\alpha$/$\beta$-LZC,
respectively. The shaded regions label the increase of potential energy as an
indication of the Zr disorder. (b) The calculated heat capacity $C_{v}$. The
first rise in $C_{v}$ corresponds to the transition to Li/vacancy disorder,
and the second rise (indicated by the shaded region) corresponds to the
transition where Zr disorders.

###  II.4 Machine-learning interatomic potentials for molecular dynamics

Recent developments of machine-learning interatomic potentials (MLIP) [47, 48]
have enabled large-scale and nano-second-long simulations to directly simulate
the Li transport kinetics near room temperature [49]. We used CHGNet to
implement MLIP-MD. CHGNet was fine-tuned with ab-initio calculations in the
generalized gradient approximation (GGA) functional [50] as described below.
The DFT calculations were performed with VASP using Gamma-point-only for
$k$-space sampling and a plane-wave energy cutoff of 500 eV. Large supercells
with cell lengths of at least 10 Å in each direction were used. The
calculations were converged to $10^{-6}$ eV in total energy and the DFT-D3
method of Grimme was used to include Van der Waals corrections [51].

The training set to fine-tune CHGNet for MD simulations was created through
the following steps. (1) Three cation configurations with different Zr
disorder were taken from MC simulations with $T_{\text{eff}}=0/3050/5000$ K
and annealed down to 0 K in MC to equilibrate the Li/vacancy configuration.
This was performed for both $\alpha$\- and $\beta$-LZC and the resulting
structures were relaxed with DFT calculations using the GGA-D3 functional. (2)
Five different configurations were generated by applying strains ($-1\%$,
$0\%$, $+1\%$, $+2\%$, $+3\%$) to these DFT-relaxed structures. (3) We used
Bayesian force fields implemented in the Flare package to sample the atomic
configurations with on-the-fly (OTF) MD [52, 53]. The DFT calculation was
called when the uncertainty threshold is higher than
std_tolerance_factor=-0.03 in Flare. OTF-MD in the NVT ensemble was initiated
from each strained configuration by heating from 0 K to the target
temperatures ($T=300/400/600/800$ K) using the Berendsen thermostat [54]. For
each MC configuration, a total of 20 OTF-MD simulations were conducted in
parallel, covering five different strains at four distinct temperatures. (4)
After the OTF-MD sampling, we fitted a trial CHGNet to initiate NPT-MD
simulations of up to 50 ps starting from the DFT-relaxed structures. The MD
trajectories were randomly sampled and the energy of these configurations,
calculated with static DFT, was added to the MLIP training set (approximately
10,000 structures). In total, over 18,000 atomic configurations and their DFT
energy were used to fine-tune the pretrained CHGNet for MD production. The
final CHGNet model (for MD simulations) achieves mean absolute errors of 1
meV/atom for energy, 34 meV/Å for interatomic forces, and 0.021 GPa for stress
on the validation set.

The MD simulations to obtain Li diffusivity were initiated from the DFT-
relaxed structures with a 50-ps initial equilibration in the NPT ensemble. The
output of these short simulations was used to initiate longer MDs in the NVT
ensemble for production. The Li self-diffusion coefficients were obtained by
calculating the mean squared displacements of Li trajectories for at least 2
ns. The diffusivity and activation energy were computed following the
empirical error estimation scheme proposed by He  _et al._ [55]. All the MD
simulations were implemented using the Atomic Simulation Environment (ASE)
interface [56].

Figure 3: (a)-(b) The Li/vacancy configurational energy landscape
$\\{E_{T}\\}$ sampled by MC annealing with fixed Zr ordering. The color lines
represent the average energy sampled by the Li/vacancy degrees of freedom with
various Zr orderings obtained by equilibration at $T_{\text{eff}}$. The rise
in $\\{E_{T}\\}$ indicates the activation of Li/vacancy inter-site exchanges
(disorder). As the $T_{\text{eff}}$ at which the Zr was disordered increases,
Li can more easily sample higher energy states. (c)-(d) Arrhenius plots of the
MD simulated diffusivities vs. temperature from 300 K to 700 K. The MD
simulations started from the Zr configurations obtained with MC at
$T_{\text{eff}}=0$ K (green triangles), 3050 K (orange dots), and 5000 K (red
squares). The dashed lines represent the activation energy fitted in the low-
temperature range ($T<500$ K). A significant non-Arrhenius behavior is
observed in Li diffusion in the ground-state structure. The presence of Zr
disorder reduces the degree of non-Arrhenius behavior, and the dashed line
fitted in low-temperature regions extends well to the high-temperature region.

##  III Results

###  III.1 Cation disorder at elevated temperatures

To investigate the thermodynamics of cation disorder in LZCs, we conducted CE-
MC simulations to sample the average and fluctuation of the configurational
energy at finite temperatures. Figure 2(a) presents the average potential
energy above the ground state $\Delta
E_{\text{pot}}=E(T_{\text{eff}})-E_{\text{GS}}$, and Fig. 2(b) shows the heat
capacity $C_{v}=\sigma^{2}(E)/k_{B}T^{2}$, where the green/orange dots
represent $\alpha/\beta$-LZC, respectively.

The two significant increases in $\Delta E_{\text{pot}}$ accompanied by peak
signals in $C_{v}$ for both $\alpha/\beta$-LZC indicate order-disorder phase
transitions. The heat capacity peak near 500 K is associated exclusively with
Li/vacancy disorder, and no Zr disorder is detected in the MC structures in
this temperature range. The subsequent increase in $\Delta E_{\text{pot}}$ and
corresponding peak in $C_{v}$ near 4000 K corresponds to the Zr order–disorder
phase transition, which is marked by the green/orange shaded region in Fig. 2.
Such a high temperature suggests that there is no spontaneous Zr disorder from
thermal fluctuations at the typical temperatures accessible to solid-state
synthesis. Given these results, it is clear that only high-energy, non-
equilibrium synthesis approaches such as mechanochemical ball-milling can
achieve the disordered state.

###  III.2 Li-site disorder conditioned on Zr orderings

To evaluate how the Li/vacancy site distribution is affected by the presence
of Zr disorder, we investigated the energy sampled by the Li/vacancy
configurations in a MC simulation when Zr ordering is fixed to that obtained
at elevated temperatures. Figure 3 (a)-(b) shows the average configurational
energy, $\\{E_{T}\\}$, sampled by the Li/vacancy degrees of freedom
conditioned on various fixed Zr states of disorder (represented by
$T_{\text{eff}}$, which is the temperature at which the Zr configurations were
obtained). The $x$-axis represents the temperature at which the Li/vacancy
energy was sampled in MC, and the $y$-axis shows the average energy difference
above the 0 K configuration. The Li/vacancy energetics for both $\alpha$\- and
$\beta$-LZC with $T_{\text{eff}}=50$ K and 1050 K are almost the same,
indicating no Zr disorder occurs in this temperature range. When Zr is well
ordered ($T_{\text{eff}}=50$ K or $T_{\text{eff}}=1050$ K), the Li/vacancy
disorder initiates at approximately 500 K indicated by the characteristic
potential-energy increase.

For the structures with a high degree of Zr disorder (e.g.,
$T_{\text{eff}}=5000$ K), Li already samples higher-energy configurations at
low temperatures (200 to 300 K) and no distinct order–disorder transition is
observed, consistent with findings in other materials with coupled disorder
[57]. This observation indicates that Li/vacancy configurational disorder at
room temperature is made possible by Zr disorder.

Figure 4: The room-temperature Li diffusion trajectories from the MD
simulations in (a)-(c) $\alpha$-LZC and (d)-(f) $\beta$-LZC. The different
degrees of Zr disorder are indicated by the effective temperatures with which
they are obtained in MC.

###  III.3 Li transport as a function of Zr disorder

To obtain quantitative measures of Li diffusivity as a function of Zr
disorder, we used MLIPs to run large-scale and long-time MD simulations.
Structures with partial Zr disorder were obtained from MC simulations at
$T_{\text{eff}}=0/3050/5000$ K. In the MD simulations, Zr ions do not migrate
so Li diffusion can be studied for a fixed Zr disorder. Figure 3(c)-(d)
displays the Li diffusivities at temperatures from $T=300$ K to 700 K in these
structures. The ground-state structures (no Zr disorder) are represented by
red squares, structures with partial Zr disorder ($T_{\text{eff}}=3050$ K) are
represented by orange dots, and structures beyond the Zr order-disorder
transition ($T_{\text{eff}}=5000$ K) are represented by green triangles. The
dashed lines in Fig. 3(c)-(d) are fits to the activation energies using the
diffusivities obtained from the low-temperature range ($<500$ K). The unfilled
squares represent diffusivities for which a limited amount of Li hopping is
observed in the MD simulations leading to high error bars on the diffusivity
(see Fig. 4(a) and (d) as examples).

For structures beyond the Zr order–disorder transition ($T_{\text{eff}}=5000$
K, green triangles), the dashed line fit from the low-temperature region
closely aligns with the high-temperature simulated diffusivities (500 K to 800
K), which indicates that the diffusion coefficient satisfies a simple
Arrhenius law over the full temperature region. In contrast, structures
without Zr disorder (ground state, red squares) exhibit marked non-Arrhenius
behavior for both $\alpha$\- and $\beta$-LZC, as they show no overlap between
the dashed lines and high-temperature diffusivities. The steep slope of these
red dashed lines indicates high activation energies near room temperature
($E_{a}=0.97\pm 0.08$ eV for $\alpha$-LZC and $E_{a}=0.68\pm 0.04$ eV for
$\beta$-LZC), significantly larger than those determined by nudged-elastic
band (NEB) calculations within the hcp and ccp halide framework [12]. The
analysis derived from MD simulations aligns closely with the thermodynamic
data. The non-Arrhenius behavior of Li transport in the structures with
ground-state Zr ordering arises from the order–disorder transition of Li and
vacancies near 500 K (see blue lines in Fig. 3(a)-(b)), whereas such a
transition is not observed for Zr-disordered structures.

The $\alpha$\- and $\beta$-LZC polymorph show distinct behavior when Zr is
partially disordered ($T_{\text{eff}}=3050$ K, orange dots in Fig. 3(c)-(d)).
The diffusivity of Li in $\alpha$-LZC with $T_{\text{eff}}=3050$ K has an
activation energy $E_{a}=0.42\pm 0.02$ eV, which is in between that of the
ground state ordering and fully disordered Zr. In contrast, for $\beta$-LZC,
Li diffusion for partial Zr disorder ($T_{\text{eff}}=3050$ K) occurs with a
similar activation energy than in the highly disordered case ($E_{a}=0.31\pm
0.02$ eV), suggesting that the ccp structures are conducive to Li diffusion
even when Zr is only partially disordered. We find that the difference of
activation energy in $\alpha$ and $\beta$-LZC is also reflected in the energy
landscape, with $\alpha$-LZC showing some flattening of the energy landscape
(orange line in Fig. 3(a)). This mirrors the higher activation energy
($E_{a}=0.42$ eV) as compared to the fully Zr-disordered state ($E_{a}=0.28$
eV), as shown in Fig. 3(c). In contrast, the energy landscape for $\beta$-LZC
(orange line in Fig. 3(b)) closely resembles that of fully-disordered states
(dark-red lines in Fig. 3(b)), consistent with the MD findings that suggest
similar activation energies ($E_{a}=0.31$ eV) for the two degrees of Zr
disorder in $\beta$-LZC.

An illustration of the difference in the Li-transport kinetics is demonstrated
in Fig. 4, where the Li diffusion paths in MD simulations are plotted. Very
limited movement of Li ions is observed within the Zr ground state structures,
as the Li-ions oscillate around their equilibrium sites (see Fig. 4(a) and
(c)). At $T_{\text{eff}}=3050$ K, limited Li hops to nearest sites are
observed for $\alpha$-LZC due to the partial Zr disorder, whereas $\beta$-LZC
shows more extensive Li diffusion. With substantial Zr disorder
($T_{\text{eff}}=5000$ K), the Li trajectories demonstrate interconnected
diffusion pathways across Li sites, which rationalize the high Li mobility
promoted by Zr disorder.

##  IV Discussion

In the realm of ASSBs, chloride-based lithium superionic conductors stand out
because of their high ionic conductivity, interface stability, and oxidation
resistance. Optimizing the Li migration barriers and increasing Li site
connectivity are important roads for enhancing Li-ion conductivity in
chloride-based solid-state electrolytes. Prior research has indicated
relatively low energy barriers ($<$ 0.3 eV) for lithium diffusion in both hcp
and ccp frameworks of chlorides from either NEB calculations or by fitting the
diffusivities derived from AIMD simulations at high temperatures [12].
However, the LZC synthesized by solid-state heating exhibits a much higher
activation energy and lower Li-ion conductivity at room temperature than would
be expected from these basic models. Ball-milled LZC, on the other hand, is
more conductive.

We selected the LZC system as a representative model to elucidate the effect
of cation disorder on ionic transport because it exists in both $\alpha$\-
(hcp) and $\beta$\- (fcc) phases, encompassing the typical anion arrangements
common for closed-packed halide conductors. We quantified the degree of cation
disorder within LZC by distinguishing between two key degrees of freedom:
Li/vacancy disorder and Zr disorder. Our CE-MC simulations reveal three key
findings: (1) no spontaneous Zr disorder is attainable through thermal
fluctuations alone at any temperature reasonably accessible in solid-state
synthesis, (2) thermal fluctuations at room temperature are inadequate to
sustain Li/vacancy disorder when Zr is well-ordered in its ground state, and
(3) Zr disorder frustrates the configurational energy landscape and enables
facile Li/vacancy disorder at room temperature. These findings are supported
by MLIP-MD (kinetic simulations), which reveal significant non-Arrhenius
behavior of the Li diffusivity when Zr is well-ordered in contrast to the Zr-
disordered states, which show typical Arrhenius behavior in the full
temperature range. Using MLIP-MD, evidence for non-Arrhenius behavior has also
been shown for several other chloride compounds, such as Li3YCl6 [26] and
Na3-xY1-xZrxCl6 [58, 59], leading to reduced conductivities near room
temperature.

Our simulations reveal that the $\beta$-LZC polymorph exhibits lower
activation energy ($E_{a}=0.68$ eV) than $\alpha$-LZC ($E_{a}=0.97$ eV) when
Zr is fully ordered, and that $\beta$-LZC achieves an activation energy for
Li-ion diffusion ($E_{a}=0.31$ eV) similar to that of the highly disordered
state even with partial Zr disorder. This suggests that the $\beta$-LZC
structure type may serve as a better framework for achieving fast Li transport
without necessitating complete cation disorder through high-energy ball
milling. This is consistent with the experimental findings that doping
$\beta$-LZC with Sc/In leading to partial cation occupancy, even via solid-
state heating, can achieve Li-ion conductivities over $2$ mS/cm [19, 20]

In summary, our study identifies metal disorder as a critical determinant for
achieving high Li transport in halide superionic conductors. In Li2ZrCl6, the
high effective temperature required for Zr disorder limits equilibrium
pathways for its synthesis and highlights the importance of manipulating
cation disorder through non-equilibrium synthesis strategies. As many such
non-equilibrium synthesis approaches are often more expensive and less
scalable, in particular given the air sensitivity of these materials, novel
strategies, including structural or chemical modifications may be required.
The application of doping strategies or the exploration of high-entropy
approaches may serve as viable pathways to introduce cation disorder, thereby
promoting the configurational entropy to facilitate the Li/vacancy disorder
[60]. Finally, given the importance of metal disorder for Li transport, it may
be required to better understand if such disorder remains thermally stable
over the typical lifetimes expected for battery materials.

##  V Acknowledgments

This work was supported by the Assistant Secretary for Energy Efficiency and
Renewable Energy, Vehicle Technologies Office, under the Advanced Battery
Materials Research (BMR) Program, of the U.S. Department of Energy under
Contract No. DE-AC02-05CH11231. The computational modeling in this work was
supported by the computational resources provided by the Extreme Science and
Engineering Discovery Environment (XSEDE), supported by National Science
Foundation grant number ACI1053575; the National Energy Research Scientific
Computing Center (NERSC); and the National Renewable Energy Laboratory (NREL)
clusters under ahlssic allocation. The authors thank Xiaochen Yang for
valuable discussions and Yu Xie for the help on the Flare package setups.

## References

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miRNA-subcellular location | located in | RNALocate | 2,572 | 15 | 3,496
snRNA-subcellular location | located in | RNALocate | 1,719 | 4 | 1,719
ncRNA-subcellular location | located in | RNALocate | 85 | 9 | 107
pseudogene-subcellular location | located in | RNALocate | 218 | 6 | 230
snoRNA-subcellular location | located in | RNALocate | 570 | 7 | 763
scaRNA-subcellular location | located in | RNALocate | 1 | 1 | 1
rRNA-subcellular location | located in | RNALocate | 2 | 2 | 3
lncRNA-subcellular location | located in | RNALocate | 1,047 | 11 | 2,313
lincRNA-subcellular location | located in | RNALocate | 22 | 5 | 28
othersRNA-subcellular location | located in | RNALocate | 3 | 1 | 3
tRNA-subcellular location | located in | RNALocate | 1 | 1 | 1
circRNA-subcellular location | located in | RNALocate | 1 | 1 | 1
Y RNA-subcellular location | located in | RNALocate | 4 | 1 | 4
scRNA-subcellular location | located in | RNALocate | 2 | 1 | 2
mRNA-subcellular location | located in | RNALocate | 13,646 | 19 | 50,903
vRNA-subcellular location | located in | RNALocate | 3 | 2 | 6
mtRNA-subcellular location | located in | RNALocate | 10 | 2 | 19
miRNA-programmed death | participates in | ncRDeathDB | 581 | 3 | 792
lncRNA-programmed death | participates in | ncRDeathDB | 99 | 2 | 99
snoRNA-programmed death | participates in | ncRDeathDB | 1 | 1 | 1
gRNA-gene | decreases by repression quantity of | Addgene | 77 | 43 | 77
ASO-mRNA | represses expression of | eSkip-Finder | 2,633 | 16 | 2,678
ASO drug-mRNA | involved in negative regulation of | DrugBank | 6 | 4 | 6
tRNA-mRNA | interacts with | RNAInter | 4 | 4 | 4
tRNA-amino acid | molecularly interacts with | tRNAdb | 549 | 19 | 549
tRNA-lncRNA | interacts with | RNAInter | 8 | 4 | 9
Supplementary Table 6: RNA-KG Descriptive statistics by primary edge type (Part V). Node type | Full name | Identifier(s)
---|---|---
Anatomy | - |  Uberon (http://purl.obolibrary.org/obo/UBERON_0005253)
Amino acid | - |  ChEBI (http://purl.obolibrary.org/obo/CHEBI_25017)
ASO | AntiSense Oligonucleotide |  Oligo name in literature (https://eskip-finder.org?H45_1-13_18-30)
ASO drug | AntiSense Oligonucleotide drug |  DrugBank (https://go.drugbank.com/drugs/DB05528)
Aptamer | RNA aptamer |  Apta-Index (https://www.aptagen.com/aptamer-details/?id=608)
Aptamer drug | RNA aptamer drug |  DrugBank (https://go.drugbank.com/drugs/DB04932)
Bacterial strain | - |  NCBI Taxonomy Browser (https://www.ncbi.nlm.nih.gov/Taxonomy/Browser/wwwtax.cgi?id=485)
Biological context | - |  |  Uberon (http://purl.obolibrary.org/obo/UBERON_0000479)
---
GO (http://purl.obolibrary.org/obo/GO_0070062)
CLO (http://purl.obolibrary.org/obo/CLO_0009828)
Mondo (http://purl.obolibrary.org/obo/MONDO_0005108)
Biological process | - |  GO (http://purl.obolibrary.org/obo/GO_0044848)
Biological role | - |  NIH Talking Glossary of Genetic Terms (https://www.genome.gov/genetics-glossary/Oncogene)
Cell | - |  CLO (http://purl.obolibrary.org/obo/CLO_0009828)
Cellular component | - |  GO (http://purl.obolibrary.org/obo/GO_0110165)
Chemical | - |  ChEBI (http://purl.obolibrary.org/obo/CHEBI_77062)
circRNA | Circular RNA |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/6575?circRNA)
Disease | - |  Mondo (http://purl.obolibrary.org/obo/MONDO_0004971)
Epigenetic modification | - |  |  GO (http://purl.obolibrary.org/obo/GO_0006306)
---
ENCODE (https://www.encodeproject.org/targets/H3K4me2)
eRNA | RNA enhancer |  Human enhancer RNA Atlas (HeRA) (https://hanlab.uth.edu/HeRA?IL1beta-RBT46)
Extracellular vesicle | - |  GO (http://purl.obolibrary.org/obo/GO_1990742)
Gene | - |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/1954)
Genomic sequence | - |  SO (http://purl.obolibrary.org/obo/SO_0000704)
gRNA | Guide RNA |  Addgene (https://www.addgene.org/41818)
Histone modification | - |  dbEM (http://crdd.osdd.net/raghava/dbem?H3K9me2)
lincRNA | Long intergenic RNA |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/100287569?lincRNA)
Lipid | - |  ChEBI (http://purl.obolibrary.org/obo/CHEBI_136143)
lncRNA | Long non-coding RNA |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/100506207?lncRNA)
miscRNA | Miscellaneous RNA |  NCBI Entrez gene (www.ncbi.nlm.nih.gov/gene/6029?misc_RNA)
mRNA | Messenger RNA |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/1756?mRNA)
mRNA vaccine | - |  DrugBank (https://go.drugbank.com/drugs/DB15654)
miRNA | Mature microRNA |  miRBase (https://www.mirbase.org/cgi-bin/mature.pl?mature_acc=MIMAT0022711)
mtRNA | Mitochondrial RNA |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/4549?mtRNA)
ncRNA | Non-coding RNA |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/102723629?ncRNA)
Other (not classified yet) RNA | - |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/3537?other)
Pathway | - |  PW (http://purl.obolibrary.org/obo/PW_0000632)
Phenotype | - |  HPO (http://purl.obolibrary.org/obo/HP_0005506)
piRNA | Piwi-interacting RNA |  piRBase (http://bigdata.ibp.ac.cn/piRBase?piR-39980)
premiRNA | Hairpin microRNA |  miRBase (https://www.mirbase.org/cgi-bin/mirna_entry.pl?acc=MI0000067)
Programmed cell death | - |  GO (http://purl.obolibrary.org/obo/GO_0097300)
Protein | - |  PRO (http://purl.obolibrary.org/obo/PR_Q92506)
Pseudogene | - |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/442240?pseudo)
Retained intron | - |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/23642?retained_intron)
RBP | RNA-Binding Protein |  PRO (http://purl.obolibrary.org/obo/PR_Q92506)
Riboswitch | - |  TBDB (https://tbdb.io/tboxes/UQCD7JYG.html)
Ribozyme | RIBOnucleic acid enZYME |  Rfam (http://rfamlive.xfam.org/family/RF02682)
rRNA | Ribosomal RNA |  snoDB (http://scottgroup.med.usherbrooke.ca/snoDB/28S-3616?snoDBrRNA)
scRNA | Small conditional RNA |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/618?scRNA)
scaRNA | Small Cajal body-specific RNA |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/677767?scaRNA)
shRNA | Short/small hairpin RNA |  ICBP siRNA (http://web.mit.edu/sirna/sequences/results-1107.html)
siRNA | Short interfering RNA |  ICBP siRNA (http://web.mit.edu/sirna/sequences/results-1053.html)
siRNA drug | Short interfering RNA drug |  DrugBank (https://go.drugbank.com/drugs/DB15935)
Small protein | - |  SmProt (http://bigdata.ibp.ac.cn/SmProt/SmProt.php?ID=SPROHSA53815)
snRNA | Small nuclear RNA |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/26824?snRNA)
snoRNA | Small nucleolar RNA |  NCBI Entrez gene (http://www.ncbi.nlm.nih.gov/gene/727676?snoRNA)
Subcellular location | - |  GO (http://purl.obolibrary.org/obo/GO_0005840)
TEC | To be Experimentally Confirmed RNA |  NCBI Entrez gene (https://www.ncbi.nlm.nih.gov/gene/8123?TEC)
TF | Transcription Factor |  PRO (http://purl.obolibrary.org/obo/PR_000007035)
tRF | tRNA-derived fragment |  |  tRFdb (http://genome.bioch.virginia.edu/trfdb?tRF-3018b)
---
MINTbase (https://cm.jefferson.edu/MINTbase/InputController?v=f&g=GRCh37&e=1&
search=submit&t=All&am=All&an=All&da=&tn=&fs=&fn=tRF-16-4YDRFUE)
tRNA | Transfer RNA |  |  GtRNAdb (http://gtrnadb.ucsc.edu/genomes/eukaryota/Hsapi19/genes/tRNA-Gln-CTG-1-3.html)
---
tRFdb (http://genome.bioch.virginia.edu/trfdb?chr8.trna4-TyrGTA)
NCBI Entrez gene (https://www.ncbi.nlm.nih.gov/gene/4567?tRNA)
tsRNA | tRNA-derived small RNA |  tsRFun (https://rna.sysu.edu.cn/tsRFun/searchDetail-tsRNA.php?tsRNAid=tsRNA-Gly-i-0605)
Unknown RNA | - |  NCBI Entrez gene (https://www.ncbi.nlm.nih.gov/gene/100128998?unknown)
Vaccine | - |  VO (http://purl.obolibrary.org/obo/VO_0000186)
Variant (SNP) | - |  NCBI dbSNP (https://www.ncbi.nlm.nih.gov/snp/rs71354105)
Viral miRNA | - |  miRBase (https://www.mirbase.org/cgi-bin/mature.pl?mature_acc=MIMAT0001581)
Viral mRNA | - |  NCBI Entrez gene (https://www.ncbi.nlm.nih.gov/gene/43740578?viral_mRNA)
Viral protein | - |  PRO (http://purl.obolibrary.org/obo/PR_000036828)
Viral RNA | - |  NCBI Nucleotide database (https://www.ncbi.nlm.nih.gov/nuccore/KF869252.1)
vRNA | Vault RNA |  NCBI Entrez gene (https://www.ncbi.nlm.nih.gov/gene/56664?vRNA)
Y RNA | - |  NCBI Entrez gene (https://www.ncbi.nlm.nih.gov/gene/6090?Y_RNA)
Supplementary Table 7: RNA-KG primary node types and their corresponding
identifiers with an instance sample.

Supplementary Listing 1: SPARQL query to retrieve all miRNA molecules that
causes or contributes to the development of leukemia.

⬇

PREFIX rdfs: <http://www.w3.org/2000/01/rdf-schema#>

PREFIX obo: <http://purl.obolibrary.org/obo/>

SELECT ?miRNA

WHERE {

?miRNA rdfs:subClassOf obo:SO_0000276. # SO_0000276 ->label-> miRNA

?miRNA obo:RO_0003302 obo:MONDO_0005059.

# RO_0003302 ->label-> causes or contributes to condition;

# MONDO_0005059 ->label-> leukemia

}

Supplementary Listing 2: SPARQL query to retrieve all premiRNA molecules that
develop into mature miRNAs known to be located in an apoptotic body and cause
or contribute to the development of a cancer.

⬇

PREFIX rdfs: <http://www.w3.org/2000/01/rdf-schema#>

PREFIX obo: <http://purl.obolibrary.org/obo/>

SELECT ?premiRNA ?disease

WHERE {

?premiRNA rdfs:subClassOf obo:SO_0000647; # SO_0000647 ->label-> premiRNA

obo:RO_0002203 ?miRNA. # RO_0002203 ->label-> develops into

?miRNA rdfs:subClassOf obo:SO_0000276;

obo:RO_0003302 ?disease.

?miRNA obo:RO_0001025 obo:GO_0097189.

# RO_0001025 ->label-> located in; GO_0097189 ->label-> apoptotic body

?disease rdfs:subClassOf obo:MONDO_0004992. # MONDO_0004992 ->label-> cancer

}

Supplementary Listing 3: SPARQL query to retrieve all lncRNA molecules that
are over-expressed in a viral infectious disease and are known to cause or
contribute to a disease treated by at least one RNA drug.

⬇

PREFIX rdfs: <http://www.w3.org/2000/01/rdf-schema#>

PREFIX obo: <http://purl.obolibrary.org/obo/>

PREFIX disease: <http://purl.obolibrary.org/obo/MONDO_>

PREFIX RNAdrug: <https://go.drugbank.com/drugs/>

SELECT ?lncRNA ?disease ?RNAdrug (COUNT(DISTINCT ?RNAdrug) as ?numRNAdrugs)

WHERE {

?lncRNA rdfs:subClassOf obo:SO_0001877; # SO_0001877 ->label-> lncRNA

obo:RO_0002245 obo:MONDO_0005108;

# MONDO_0005108 ->label-> viral infectious disease

obo:RO_0003302 ?disease.

?disease obo:RO_0002302 ?RNAdrug.

# RO_0002302 ->label-> is treated by substance

FILTER(STRSTARTS(STR(?disease), STR(disease:)))

FILTER(STRSTARTS(STR(?RNAdrug), STR(RNAdrug:)))

}

GROUP BY ?lncRNA ?disease ?RNAdrug

HAVING (COUNT(DISTINCT ?RNAdrug) >= 1)