Path entropy-driven design of solid-state electrolytes

Domain: biology
Published: 2026-04-01
Editorial outcome: major_revision

Abstract
The development of high-performance solid-state electrolytes (SSEs) has entered a critical stage, where entropy-driven strategies offer transformative potential for enhancing electrochemical properties. By engineering local environments for conductive ions alongside introducing disorder, these approaches can significantly improve conductivity. However, embracing high-entropy designs does not always guarantee improved performance. Current entropy descriptions oversimplify disorder by accounting solely for host framework configurations, neglecting conductive ion-induced disorder, rendering such descriptions incomplete. Herein, we propose path entropy (S ) as a descriptor that quantifies diffusion pathway diversity, directly capturing diffusional disorder. Combining Markov state model with transition path theory, we reveal the interplay between diffusion pathway diversity of lithium and microscopic local environments in inorganic thiophosphates. Generalizing this path-informative S for high-throughput screening, we demonstrate its broad applicability in identifying and designing high-performance SSEs. Our work establishes a critical link between entropy evolution underlying ion conduction and practical entropy-driven design principles.

Introduction
Recent years have underscored the critical role of high-performance batteries in storing intermittent renewable energy1. Driven by the pursuit of all-solid-state metal batteries characterized by high energy density and safety, SSEs with high ionic conductivity have received substantial attention2,3. However, the inclusion of many-body and group-concerted motions in ionic conduction results in significantly more complex behavior in SSEs than in single-entity or classical ionic conductors3-5, rendering dramatic challenges in theoretical and experimental design. In fact, experimentally verified bond breaking and reorganization triggered by ion conduction occur at frequencies up to 40 THz6, revealing ultrafast fluctuations of thermodynamic variables. A rigorous thermodynamic formulation understanding ion conduction incorporates the free energy barrier (∆𝐺), enthalpy change (∆𝐻), and entropy change (∆𝑆) at a specific temperature 𝑇: ∆𝐺 = ∆𝐻 −𝑇∆𝑆. Consequently, minimization of the ∆𝐺 can be counteracted by an increase in the ∆𝑆, underpinning entropy-driven strategies. By engineering local environments while introducing disorder, ionic conductivity in a specific system can be enhanced through methods such as doping7,8, vacancy creation9-11, anion substitution12-15, etc. While the enthalpies associated with bond formation and breaking are relatively straightforward to quantify, the entropy changes accompanying ionic diffusion remain challenging to characterize. To date, most entropy descriptions in ionic systems focus on structural disorder via configurational entropy, which is often derived from high-entropy alloys12,16,17. Alternative descriptors, such as “migration entropy”18,19, depend solely on lithium-ion hopping/vibrational frequencies. The pathway of diffusive ions has been largely overlooked, leaving the entropy originating from ionic diffusion in SSEs unresolved.

The core challenge lies in quantitatively assessing the entropy arising from all parts in SSEs. Focusing solely on site occupancy information from configurational entropy or metrics such as hopping frequencies18 and mean square displacement20 is insufficient. Analysis of diffusion pathways from one site to another, where ions spend a significant fraction of time21, is fundamental. The diversity of these diffusion pathways is therefore central in quantifying entropy generated by ionic conduction. Inspired by principles of information theory applied to thermodynamics22,23, we propose that entropy arising from ionic diffusion can be directly quantified through path entropy (S ), a metric that captures information about diffusion pathways traversed by lithium ions. Similar to how configurational entropy encodes structural disorder, path entropy encodes diffusional disorder, which is determined by the routes ions take during their motion. (Fig. 1)

Here, we combine the Markov state model22 and transition path theory24,25 to establish a systematic approach for quantifying disorder in SSEs. By decomposing configurational and diffusional entropy contributions, we establish a critical link between local environment engineering and enhanced ionic diffusion. As a proof-of-concept, we employ Li-argyrodite SSEs as model systems. We assess the entropy-driven strategies, such as vacancy creation and anion substitution in Li-argyrodites, through both diffusional and configurational disorder. Critically, conceptions of interpreting the disorder as solely configurational disorder can be misleading, as some strategies (e.g., lithium vacancy creation) introduce minimal configurational disorder but yield substantial improvements in ionic conduction. Taking account of diffusional disorder quantified through path entropy (S ) is therefore crucial as it directly reflects the diffusion capacity of lithium ions. Generalizing this entropy-analysis protocol across a broader range of inorganic sulfide SSEs via high-throughput screening, we identify Li Cr C SO (with ionic conductivity up to 5.05 ± 0.23 mS/cm),which exhibits performance comparable to the well-established argyrodite SSEs.

Results
Entropy-driven engineering of argyrodite-type SSEs

Among various SSE families, including halides, oxides, and polymers, sulfides are particularly notable for their high ionic conductivity (σ), typically ranging from 10-5 to 10-2 S/cm14. Li-argyrodites (Li BC D , 0 ≤ x ≤ 1, B = P or As; C = S or Se; D = Cl, Br or I), a prominent subtype within the sulfide family with rich polyanionic moieties26, are distinguished by their facile synthesis process and high σ27. In Li-argyrodites, the rigid anion framework [PS ]3- serves as the structural backbone, accommodating the flexible lithium coordination shells26,28. To explore entropy-driven approaches for modulating anion framework – lithium coordination shell interactions, we selected four argyrodite-type SSEs candidates (Fig. 2a-d).

We chose the Li-P-S-Cl system as our model, which includes Li P(S Cl) and Li PS Cl with 𝐴𝑚𝑚2 and 𝐹4̅3𝑚 space groups, designated as LPSCl-I and LPSCl-II (experimentally confirmed superionic phase29), respectively. Here, LPSCl-II serves as the benchmark superionic phase, while LPSCl-I acts as a non-superionic reference. To tailor local environments of LPSCl-II via entropy-driven engineering, we implemented two strategies: (i) introducing lithium vacancies and site disorder (sulfur and chloride ions occupying Wyckoff sites 4a and 4c randomly) to form LPSCl-III phase (Li PS Cl 9); and (ii) substituting anion in the framework (replacing P with Si) to generate a new phase as LSPSCl (Li Si P S Cl14,30). The atomic structure of LPSCl-III was determined using cluster expansion, guided by experimental data9 (Supplementary Fig. 1-3), while LSPSCl was sourced from Materials Project30,31.

Diffusion coefficients (𝐷) and the σ were calculated (Supplementary Note 1, Supplementary Fig. 4 and 5). Both engineered phases, LPSCl-III and LSPSCl, exhibit enhanced ionic conduction. At 300 K, LPSCl-III and LSPSCl achieve diffusion coefficients of 2.68 × 10-11 ± 1.52 × 10-13 m2/s and 7.22 × 10-12 ± 1.95×10-13 m2/s, at least three orders of magnitude higher than pristine LPSCl-II (4.24 × 10-15 ± 1.98×10-16 m2/s) (Supplementary Table 1). Ionic conductivities calculated via the Nernst-Einstein equation align closely with experimental measurements: the σ values for LPSCl-II, LSPSCl, and LPSCl-III are 1.53 × 10-3 ± 1.40 × 10-4 mS/cm, (consistent with experimental values32,33 around 10-3 mS/cm), 2.37 ± 0.13 mS/cm, and 8.42 ± 0.093 mS/cm (close to reported value9 of 9.4 mS/cm), respectively. LPSCl-I manifests the lowest conductivity (1.10 × 10-7 ± 3.82 × 10-8 mS/cm), confirming its role as a non-superionic reference.

Modeling lithium-ion diffusion through Markov state models

To capture collective lithium-ion dynamics relevant to battery operation in these engineered SSEs, we employ Markov state models (MSMs)24,25, an effective way for modeling SSE systems34,35. Lithium-ion diffusion in SSEs proceeds via discrete site occupancy/vacancy transitions, forming a continuous-time Markov chain within a discrete state space. To ensure sufficient transitions (a common challenge in modeling Markov chains25), we utilize neural network potential-based molecular dynamics (NNMD) simulations trained at the density functional theory (DFT) level (Supplementary Note 2, Supplementary Tables 2-3, Supplementary Fig. 6-11). This enables large-scale molecular dynamics simulations spanning nanometer length scales and nanosecond time scales for our four systems containing thousands of atoms (Supplementary Fig. 12). These trajectories were then used to construct the MSMs.

We first identify discrete lithium-ion spaces via local coordination shells. Ions with proximal positions exhibit similar behavior due to analogous coordination environments, which we define as local coordination shells (LCSs). Following a periodic K-means clustering method (Supplementary Note 3), which determines the optimal number of LCSs, lithium ions are assigned to LCSs characterized by distinct spatial and angular distributions. As shown in Fig. 2e-h, three LCS types are identified in LPSCl-I, four types in LPSCl-II and LPSCl-III, and five types in LSPSCl. We then discretize each LCS into partitioned 3D subspaces representing discrete lithium states (LSs) using classical Voronoi partitioning (Fig. 2i, details in partitioned boundaries and error estimation are provided in Supplementary Note 3). Specifically, a total of 10 LSs are identified for LPSCl-I, 24 LSs for LPSCl-II, 22 LSs for LPSCl-III, and 20 LSs for LSPSCl. For the remaining subspaces, the LS is designated as LS , representing the possible state lithium ions occupy during diffusion. This methodology enables spatial mapping of lithium-ion trajectories into discrete states (Supplementary Fig. 13), as illustrated in Fig. 2j. After mapping into the lithium state space, we build MSMs for each type of the SSE. As demonstrated in Fig. 2k, we resolve five lithium-ion dynamic processes in the discrete space of LPSCl-I I for an individual lithium ion.

A key advantage of the MSM approach is the direct capability to extract kinetic information via mean first passage times (MFPTs)36, quantifying the average time required for transitions between LSs. The MFPT profiles that capture state transitions across LCSs enable a holistic evaluation for both short-range (intra-LCS) and long-range (inter-LCS) diffusion pathways, revealing how local environments govern lithium-ion diffusion kinetics. While ideal MFPT values require infinite sampling, nanosecond-scale DFT-level simulations effectively capture kinetic differences in ionic diffusion: inter-LCS diffusion is completely blocked in LPSCl-II (Supplementary Fig. 14-17). In contrast, substantial inter-LCS diffusions emerge in LPSCl-III (Supplementary Fig. 18-21) and LPSCl (Supplementary Fig. 22-26), enabling efficient long-range transport. This disparity reveals that tailoring local environments through lithium-vacancy introduction and S/Cl site disorder significantly accelerates inter-LCS transport. For the anion-substituted LSPSCl, while certain states are infrequently visited at room temperature, enhanced LCS diversity also facilitates lithium-ion diffusion across different locations.

Quantify diffusional disorder through path entropy

Diffusion pathways are strongly correlated with disorder induced by ionic conduction. Through transition path theory (TPT), we characterized lithium-ion flux patterns, including pathway multiplicity and associated weights, across all LSs. Lithium ions in LPSCl-I are immobile, exhibiting no diffusion pathways across LSs. This result is consistent with its lowest diffusion coefficient among all four systems examined. Local environment engineering in LPSCl-III and LSPSCl yields demonstrably greater pathway diversity compared to pristine LPSCl-II. Specifically, lithium-ion transport from LS to LS in LPSCl-II follows only several dominant pathways (Fig. 3a). In contrast, LPSCl-III and LSPSCl exhibit expanded pathway networks (Fig. 3b and c), with substantial increases in cross-LCS state transfer events originating from multiple LCSs. This enhanced pathway multiplicity directly correlates with improved ionic conductivity in these two engineered materials.

To quantitatively assess this diffusional disorder, we define path entropy 𝑆 via Shannon entropy formulation22,23:

𝑆 = −𝑘 ∑∑𝜌𝐴𝐵ln𝜌𝐴𝐵 (1)
p 𝐵 P𝑖𝑗 𝑖𝑗
𝐴≠B 𝑖≠ 𝑗

where 𝜌𝐴𝐵 represents transition probability density between two LSs, i.e. A and B, quantifying the probabilities of all diffusion pathways through intermediate LSs (∀𝑖 ∈ (𝐴∪𝐵)𝑐) (see Methods). The entropic contribution from long-range diffusion, termed escape entropy 𝑆 , is defined as transitions originating from the original LCS (𝐴 ∈ LCS) to states outside this LCS (𝐵 ∉ LCS).

As shown in Fig. 3d and Supplementary Tables 4-7, lithium-deficient LPSCl-III exhibits significantly higher 𝑆 (2598.16 ± 112.58 J/mol/K) and 𝑆 (513.44 ± 26.86 J/mol/K) compared to LPSCl-II (𝑆 = 415.72 ± 6.02 J/mol/K; 𝑆 = 0 J/mol/K). This indicates the emergence of enhanced pathway multiplicity, particularly long-range diffusion channels. Introducing additional anions via substitution in LSPSCl also diversifies flux patterns, increasing 𝑆 (553.91 ± 54.42 J/mol/K) and 𝑆 (91.70 ± 15.82 J/mol/K).

Free energy profiles for both inter-LCS and intra-LCS diffusion were constructed to verify a positive correlation between increased diffusional disorder (path entropy) and reduced diffusion barrier by well-tempered metadynamics (WTmetaD). For inter-LCS diffusion, collective variables (CVs) D1 and D2 represent the distances from the selected lithium ion to the centers of their assigned LCSs (Supplementary Note 4.1, Supplementary Fig. 27). LPSCl-III exhibits significantly lower inter-LCS diffusion barriers (~ 38 kJ/mol) compared to LPSCl-II (~ 65 kJ/mol) (Fig. 3e and f, Supplementary Fig. 28 and 29). In contrast, intra-LCS motion shows no significant reduction in barrier height in LPSCl-III relative to LPSCl-II (Supplementary Note 4.2, Supplementary Fig. 30-33). This disparity is consistent with enhanced non-stoichiometric flexibility in LPSCl-III, where the increased 𝑆 drives inter-LCS hopping.

Quantify configurational disorder via configurational entropy

The rapid rotation of the [PS ]3- moiety in argyrodite SSE systems occurs at ~ 109 s-1 37, significantly modulating lithium-ion mobility. To compute rotational free energy, we selected the polar angle (θ) and azimuthal angle (ϕ) of tilted [PS ]3- tetrahedra as two CVs. (Fig. 4a, Supplementary Note 4.3, Supplementary Table 8). As shown in Fig. 4b and Supplementary Fig. 34, LPSCl-I exhibits restricted rotation and a narrow tilting-state distribution, consistent with its low ionic conduction. In contrast, LPSCl-II (Fig. 4c and Supplementary Fig. 35) and LPSCl-III (Fig. 4d and Supplementary Fig. 36) display unlocked rotation, with LPSCl-III showing significantly lower rotational free energy across the configurational space due to the introduced lithium vacancies. Notably, LSPSCl exhibits a slightly higher rotation barrier of [PS ]3- tetrahedra than LPSCl-III (Fig. 4e and Supplementary Fig. 37), while a much higher barrier of ~ 300 kJ/mol for Si-substituted tetrahedra ([SiS ]4- or [SiS Cl]3-) (Fig. 4f and Supplementary Fig. 38-40). While elevated barriers restrict bond rotation in [SiS ]4- and [SiS Cl]3-, the introduction of Si-substituted tetrahedral species generates distinct configurations of rigid tetrahedra units, which collectively enhance lithium-ion diffusion channel versatility.

To quantify anion framework structural disorder, we examined tetrahedral distortion (𝛿 ), defined as the deviation of [PS ]3- units from ideal tetrahedral geometry. Configurational entropy 𝑆 is then calculated as38:

𝑆 = 𝑘 ln𝑊(𝑿𝛿 ) (2)
c B d

where 𝑘 is the Boltzmann constant and 𝑊(𝛿 ) quantifies accessible configurations determined by the 𝛿 distribution (see Supplementary Note 5). As shown in Fig. 4g, LPSCl-I exhibits the lowest 𝑆 of 39.41 ± 1.48 J/mol/K with no structural disorder introduced. LPSCl-II and LPSCl-III have a similar 𝑆 (42.63 ± 1.51 J/mol/K, 42.78 ± 1.50 J/mol/K, respectively), indicating that lithium vacancies or site disorder do not significantly increase structural disorder. Notably, LSPSCl has the highest 𝑆 of 47.31 ± 1.54 J/mol/K, reflecting enhanced disordering due to anion substitution.

Connection between configurational entropy and path entropy

Entropy contributions related to ionic conduction in SSEs arise from both the flexible LCSs containing mobile lithium ions and the rigid host framework that forms the structural backbone. Disorder in these systems originates from two components: diffusional disorder associated with lithium-ion motion and configurational disorder in the host lattice. Consequently, total entropy is partitioned into path entropy 𝑆 from the flexible LCSs and configurational entropy 𝑆 from the rigid host framework. By combining these two components, the system-level disorder can be quantified rather than assessed solely through the disorder induced by the lithium-ion conduction or host framework configurations.

It is noteworthy that while systems with high 𝑆 may exhibit high 𝑆 , no direct causal relationship exists between these quantities. Instead, they represent complementary facets of solid-state ionic conductors. Joint analysis of 𝑆 and 𝑆 enables unambiguous identification of entropy gains in entropy-driven-designed systems. Compared to minor variations in the 𝑆 by different entropy-driven strategies (e.g., vacancy introduction), the 𝑆 , a metric quantifying diffusional disorder, aligns strongly with ionic diffusion performance. Its values range from 0.0 J/mol/K (LPSCl-I) to 415.72 ± 6.02 J/mol/K (LPSClI-II), 553.91 ± 54.42 J/mol/K (LSPSCl), and 2598.16 ± 112.58 J/mol/K (LPSCl-III) (Fig. 4g). These results confirm that the 𝑆 provides a more direct and quantitatively robust metric for assessing ionic diffusion than the 𝑆 . A more nuanced separation of escape entropy 𝑆 from path entropy 𝑆 , enables the clear identification of long-range diffusion in LSPSCl (91.70 ± 15.82 J/mol/K) and LPSCl-III (513.44 ± 26.86 J/mol/K).

For Li-argyrodites, configurational disorder is readily quantifiable due to the shared anion framework. However, cross-system calibration remains challenging owing to divergent host frameworks. In contrast, the 𝑆 metric, dependent exclusively on pathway diversity, accounts for a major component of total disorder in ionic systems, establishing it as a universal metric applicable across diverse ionic conductors.

High-throughput screening of SSE candidates using path entropy

To rigorously validate S as a key and general indicator of promising SSE candidates, we performed high-throughput screening using the Alexandria39 and Materials Project databases31. Our initial search encompassed 2524 inorganic sulfides, excluding systems with non-lithium mobile cations (Fig. 5a). Subsequent filtering criteria included: (i) a band gap greater than 2.0 eV (𝐸 ≥ 2.0 eV) and (ii) an energy above hull (𝐸 ) ≤ 85 meV per atom (comparable to LPSCl-II). This reduced the candidate pool to 509 materials. While these candidates satisfied the initial screening criteria for structural and electronic stability, ionic mobility remains a critical factor for SSE performance. To ensure sufficient lithium mobility, we employed NNMD simulations (Supplementary Notes 6.1 and 6.2, Supplementary Fig. 41) to compute mean square displacements (MSDs) of lithium ions. Candidates with measurable lithium-ion mobility, defined as MSDs > 0.010 nm2 over 12.0 ps of NNMD simulation, were retained. This procedure narrowed the list to 27 structurally stable compounds with detectable ionic motion (Supplementary Table 9).

Finally, we implemented a final screening step using S and S metrics, which quantitatively characterize diffusion pathways diversity and elucidate relative contributions of long-range versus short-range ionic mechanisms. This step identified seven candidate materials (Fig. 5b, Supplementary Fig. 42) with the S > 200.0 J/mol/K, including experimentally verified SSEs such as argyrodite-type compounds14,29,30 (LPSCl-II (Li PS Cl), Si-substituted LSPSCl13,14 (Li Si P S Cl ), Li PS Br, and Li PS 40), thioborate-type41 Li B S 42, Li P S 43, rock-salt sulfide Li NbS 44, and LGPS-type structures45 (Li Ge(PS ) , Li Sn(PS ) ). Notably, we also identified a promising high-performance candidate: Li Cr C SO (S = 263.98 ± 10.42 J/mol/K). This material exhibits a S value close to that of Li PS Cl (273.52 ± 0.00 J/mol/K), establishing a robust performance baseline.

Fitting the Nernst-Einstein equation (Fig. 5b, Supplementary Fig. 43, Supplementary Note 6.3, Supplementary Table 10), Li Cr C SO achieves ionic conductivity of 5.05 ± 0.23 mS/cm (Supplementary Fig. 44; activation energy 0.18 eV). Notably, this conductivity rivals that of LPSCl-III (8.42 ± 0.093 mS/cm), where long-range ionic diffusion is activated via lithium-ion vacancies and site disorder. Li Cr C SO exhibits 32 lithium sites with sparse distribution compared to LPSCl-II, resulting in 5 connected LCSs (Supplementary Fig. 45). This LCS configuration enables enhanced lithium-ion transport kinetics (Supplementary Fig. 46-50), which fall between the constrained kinetics of LPSCl-II (Supplementary Fig. 14-17) and the fully activated diffusion in LPSCl-III (Supplementary Fig. 18-21). The partially unlocked out-LCS conduction facilitates higher ionic conductivity (Supplementary Fig. 44) despite comparable path entropy to LPSCl-II. The nearly-zero S (1.8 J./mol/K; Supplementary Table 9) in Li Cr C SO confirms predominantly short-range diffusion mechanisms, analogous to pristine LPSCl-II. This characteristic suggests that employing entropy-driven strategies (e.g., vacancy engineering or site disorder) shown in step 6 of Fig. 5a could further significantly enhance its superionic performance.

Methods
Ab initio molecular dynamics simulations.

First-principle calculations have been performed through CP2K software package46. The Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA)47 with the double-zeta valence polarized basis set and Goedecker–Teter–Hutter pseudopotentials were adopted48. The auxiliary plane wave basis set was truncated using a density cutoff of 500 Ry and the van der Waals (vdW) interactions were evaluated through Grimme D3 correction. A time step of 2.0 fs was used in all ab initio molecular dynamics (AIMD) and well-tempered metadynamics (WTmetaD) simulations. The Nosé-Hoover thermostat and Martyna-Tobias-Klein barostat were used with coupling constants of 1.0 ps and 0.5 ps, respectively.

Identify the crystal structure of LPSCl-III (Li PS Cl ).

The crystal structure of LPSCl-III was verified based on the experimental findings9. All elements can be classified into Wyckoff sites as follows: Li (48h), P (4b), Cl (4a and 4c), S (4a, 4c, and 16e). (see Supplementary Fig. 1) To obtain the exact structure of Li PS Cl , which includes lithium vacancies and site mixture of Cl/S at Wyckoff 4a and 4c sites, we initially employed an enumerative approach to generate possible structures consistent with the site positions. Over a range of cell sizes from 1 to 4, we generated more than 1000 initial structures through enumeration. For 340 structures, we performed first-principles calculations (Please refer to Supplementary Fig. 2). Next, we employed cluster expansion (CE) to identify the most stable candidates among the target structures (Supplementary Fig. 3). The atomic configuration of the LPSCl-III can be represented by a string of occupation variables, {ζ ,ζ ,⋅⋅⋅,ζ }, where the ζ represents the atomic species occupying the nth site in an N-site supercell. The lattice model of the atomic configuration can be written as a sum of cluster interaction functions49:

𝐻(ζ) = ∑ 𝐻 (ζ ) (3)
𝑆 𝑆
𝑆⊆[𝑁]

In our CE model, [𝑁] is the set of all site indices, and ζ is the set of all occupation variables for sites in a cluster 𝑆. The CE model includes pairs of sites separated by less than 6 Å, triples with points less than 5 Å, quadruplets 4 Å apart, and quintuplets 4 Å apart, resulting in a total of 27 correlation functions. Initially, we selected 340 possible candidates to create the CE model, achieving a final mean squared error of 100.05 meV/prim (primitive cell) in predicting the energies. Subsequently, using this model, we predicted around 1100 structures of LPSCl-III as candidates. We then filtered more than ten energetically favorable structures included for neural-network potential training.

Neural-network potential training.

The initial training datasets were prepared through AIMD simulations conducted using the CP2K software package46. These simulations were carried out for all four argyrodite-type solid-state electrolytes across temperature ranges from 300 K to 1500 K. Subsequently, neural network-based potentials (NNPs) were trained using the Deep Potential Smooth Edition (DeepPot-SE)50, which incorporates both angular and radial information from atomic configurations. The detailed active learning for NNP training is outlined as follows (Supplementary Fig. 6): (1) For each type of SSE, we conducted 10.0 ps of AIMD simulations at temperatures of 300 K, 600 K, 900 K, 1200 K, and 1500 K, and a pressure of 1 atm under the NPT ensemble. Subsequently, for each NPT simulation, we selected two frames as candidates for the next MD simulation under the NVT ensemble. Under the NVT ensemble, we employed the On-the-fly Probability Enhanced Sampling (OPES) method51 with multithermal sampling to enrich the dataset. The internal energy (U) of the system was used for defining the CV by Δu defined as

Δ𝑢 = (𝛽′−𝛽)𝑈 (4)
𝛽′

where 𝛽 = 1/(k T) and k isBoltzmann constant, 𝛽′ is the inverse thermodynamic temperature to be sampled. For the ten AIMD simulations under the NVT ensemble, we sampled temperature ranges {(0 to 600 K), (300 to 900 K), (600 to 1200 K), (900 to 1500 K)} using multithermal simulations. Each simulation was run for at least 5.0 ps under the NVT ensemble. Note for LPSCl-III, we conducted five MD simulations under the NVT ensemble for all ten candidates predicted to be stable by our CE model. (2) Training the NNP using DeepPot-SE involved extracting energies and forces from the previous MD trajectories. The detailed parameters used for training are provided in Supplementary Table 2. (3) Based on the initial NNP models, a series of NNMD simulations were conducted under both NVT and NPT ensembles using enhanced OPES multithermal-multibaric (MTB). With OPES-MTB, the simulations were biased across a range of temperatures (300 K to 1200 K) and pressures (0.90 atm to 1.10 atm). Candidate configurations were identified based on model deviations of force falling within the range of 0.15 to 0.25 eV/Å. (4) The energies and forces of these candidates were calculated using CP2K, and the resulting data were added to the initial training sets. (5) With the enriched dataset, we iteratively train the new dataset to optimize the model and improve its accuracy. This iterative training process involves updating the neural network potential model based on the new data, refining the model parameters, and repeating the training process until the desired level of accuracy (greater than 99%) is achieved.

To validate the NNPs, we compare the mean absolute errors and radial distribution functions of lithium between the results obtained from AIMD calculations and the predictions made by the NNPs. (Supplementary Table 3, Supplementary Fig. 7-11)

Markov state model construction.

The construction steps of our MSM in studying lithium-ion diffusion are as follows (Supplementary Fig. 13): MD simulations were performed under the NVT ensemble at room temperature (300 K). We then classified LCSs of each SSE structure and discretized them (Supplementary Note 3). Lithium ions in each LCS were uniquely assigned to discretized states. A lithium-ion diffusion trajectory can thus be modeled as a chain of random variables 𝐻 ,𝐻 ,…,𝐻 ,… over discrete time moments (t) with 𝐻 being randomly one state in the state space {X}. Within the framework of the Markov method, the probability distribution (p) of 𝐻 variable at the time moment 𝑡+1 is assumed to depend only on the variable at the prior moment (i.e., 𝑡), thus we have:

𝑝(𝐻 |𝐻 ,𝐻 , ,𝐻 ,𝐻 ) = 𝑝(𝐻 |𝐻 ) (5)
𝑡+1 𝑡 𝑡−1⋯ 2 1 𝑡+1 𝑡

Therefore, the time-series trajectories of lithium atoms are equivalent to a time series of probabilistic variables. The propagation of the probability density function 𝐩(𝑡) over a time interval τ (lag time) is represented as:

𝐩(𝑡+τ)𝑇 = 𝐩(𝑡)𝑇𝐓(τ) (6)

and its explicit form expressed as:

𝑛
𝐩 (𝑡+𝜏) = ∑𝐩 (𝑡)ℙ(𝑥(𝑡+𝜏) ∈ 𝑋 ∣ ∣ 𝑥(𝑡) ∈ 𝑋 ) (7)
𝑗 𝑖 𝑗 𝑖
𝑖=1

where 𝐓(𝜏) is the transition probability matrix (TPM) with its component T being the transition probability between 𝑋 and 𝑋 over 𝜏, and its eigenvectors offer insights into the population flux of the dynamic process of lithium site hopping. Consequently, its eigenvalues (λ ) denote the timescales of these dynamic processes, which can be depicted in implied time scales (ITS):

τ
ITS (τ) = − (8)
𝑖 lnλ (τ)
𝑖

where 𝑖 = 1,2,3,4,⋅⋅⋅ represents ith eigenvalue of the 𝐓(𝜏) . To ensure the reduced state space remains memoryless (or reducing history effect) in capturing the kinetics of lithium hopping, we select a lag time τ = 600 𝑠𝑡𝑒𝑝𝑠 (12.0 picosecond). Each LCS in the trajectory contributes to the construction of the MSM, helping to reduce both random and systematic errors in the model. To assess the validity and equality of the lithium hopping MSMs, the Chapman-Kolmogorov (C-K) test was employed. This test evaluates the consistency of transition probabilities between different discrete states over time (Supplementary Fig. 51-53).

Calculation of lithium flux and path entropy.

We implemented TPT to analyze the reactive trajectories of lithium-ion transition. From TPT, we analyzed the flux of the lithium ion that moves from different states52. The lithium-ion jumping process was modeled on the countable state-space 𝑆 with rate matrix 𝐿 = (𝑙 ) :

𝑙 ≥ 0 ∀𝑖,𝑗 ∈𝑆,𝑖 ≠ 𝑗
{ (9)
∑𝑙 = 0 ∀𝑖 ∈ 𝑆
𝑖𝑗
𝑗∈𝑆

where 𝑙 represents the process jump from state 𝑖 to state 𝑗. Given the initial state of lithium at 𝑖 and the final migrated state at 𝑗, TCPT can compute the reactive flux between any two nonempty, disjoint subsets (e.g., A and BT) of the state-space 𝑆. The reactive trajectory P, a set including all the ordered sequences A𝑃 generated from the 𝑛th transition between initial state A and end state B, is defined as:

𝑃 = ⋃𝑃 (10)
𝑛
𝑛∈ℤ

Then for this reactive trajectory P, the discrete forward committor 𝑞+ = (𝑞+) is defined as the probability that the process starting in 𝑖 ∈ 𝑆 will first evolve toward B rather than A. Similarly, the probability of the process arriving in state 𝑖 last came from A rather than B is the backward committor 𝑞− = (𝑞−) . We can calculate the probability current of reactive trajectories P25:

π 𝑞−𝑙 𝑞+, if 𝑖 ≠ 𝑗
𝑓𝐴𝐵 = { 𝑖 𝑖 𝑖𝑗 𝑗 (11)
𝑖𝑗
0, otherwise

where π is the unique stationary distribution. The flux between different states is conserved through:

∑(𝑓𝐴𝐵 −𝑓𝐴𝐵) = 0 ∀𝑖 ∈ (𝐴∪𝐵)𝑐 (12)
𝑖𝑗 𝑗𝑖
𝑗∈𝑆

After having transition flux matrix 𝑓𝐴𝐵, the corresponding probability density 𝜌𝐴𝐵 is calculated through L1 normalization for all the components. Ionic transfer between different states 𝑖 and 𝑗 is considered effective if the probability exceeds 0.15. The standard error of path entropy is calculated within the transfer probability range of [0.14, 0.16]. For four types of SSE, three independent simulations were performed to compute probability densities of the net transition flux and associated path entropies. The mean path entropies fall within a 99.5% confidence interval.

Free energy calculation through well-tempered metadynamics.

We utilize well-tempered metadynamics (WTmetaD) simulations to investigate lithium-ion diffusion. Various CVs are designed to capture migration behaviors in distinct local environments. For specifics on CV design (Supplementary Fig. 27 and 30) and simulation parameters (Supplementary Table 8), please refer to Supplementary Note 4.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Dear authors,\nThank you for the insightful manuscript. It is very well written and indeed the newly defined path entropy is a central\nparameter to decouple merely the configurational entropy from potential enhancement in ionic diffusion in compositionally\ncomplex or high entropy materials.\nThe manuscript can be accepted for publication after addressing the following questions:\n1. Line 79: Li4Cr2C4SO16 is a computationally identified material. What is the main limitation for its synthesis and\nexperimental validation? Please clarify.\n2. Line 149: Are these total LSs or per LCS?\n3. Line 155: How many dynamic processes were identified for the other compositions? Is this information included\nsomewhere else?\n4. Line 233: Please elaborate on the distinct rotation behaviour, if a higher barrier was calculated. Other factors previously\ncalculated like the inter-LCS diffusion seem to be more relevant for enhancing ion conduction in LSPSCl than tetrahedra\nrotation? Please discuss in more detail.\n5. Why do Sp and Se in the high-throughput screening for LSPSCl differ from previous calculations shown in Fig.4?\n6. Line 461: Why is the standard error in the same order as the probability? Or is an error of 0.01 acceptable in this case?\nPlease clarify.\n7. How does the model consider the influence of the electric field? Experimentally, long range ionic conductivity is mostly\ndetermined by applying a bias (AC). Is it, nevertheless, possible to correlate the simulated and experimental findings?\nPlease clarify in a short discussion.", "reviewer_2": "In this paper, the authors proposed a descriptor quantifying diffusion pathway diversity—path entropy (Sp), addressing the\nlimitation of traditional configurational entropy that focuses on structural disorder. Through combining Markov state models\n(MSMs) with transition path theory (TPT), this work revealed the correlation between local environments and ionic diffusion\nin solid-state electrolytes. Furthermore, this work developed a path entropy-based high-throughput screening process for\ninorganic thiophosphates, enabling efficient identification of high-performance candidate materials.\nIn conclusion, I recommend that this work can be considered to publish in Nature Communications after Major revision.\nSeveral comments and questions are included as follow:\n1. Why did this work focus merely on verifying the strategies of lithium vacancy creation and anion substitution, instead of\nexploring other entropy-driven approaches (e.g., cation doping or multicomponent integration)? By the way, investigating\nthese additional methods could further demonstrate the general applicability of pathway entropy as a design criterion.\n2. The authors defined escape entropy (Se) as “the contribution from transitions from the original LCS to external states”, but\nfailed to clarify the standard for defining the “original LCS” (i.e., how to define the boundaries of LCS). We suppose that\nrelevant numerical standards can be added.\n3. Based on the last review, the gap of LCS among LPSCl-I, II, III, and LSPSCl (3/4/4/5 types) and whether it is related to the\nintrinsic structure of the materials are expected to be explained.\n4. In the flowchart that depicting the multistep high-throughput screening for identifying promising SSE candidates (Fig.5a),\nthe way to determine the threshold setting of “MSD > 0.018 nm2” in Step 3 remains vague. Similarly, in Step 4, why “200\nJ/mol/K” was chosen as the critical value? Furthermore, is this value universal or specific for sulfide SSE systems?\n5. It is necessary for authors to give an in-depth analysis of the fundamental relationship among Sp, Se and Sc. For\nexample, several systems (e.g., LSPSCl) have both high configurational entropy and high path entropy, while others (e.g.,\nLPSCl-III) show low configurational entropy but extremely high path entropy. Is this discrepancy related to the changes in\nlithium coordination environment?\n6. For the novel candidate material Li4Cr2C4SO16, only ionic conductivity at 300 K is provided. To determine its activation\nenergy and high-temperature stability, the temperature dependant EIS (e.g., Arrhenius plot) is suggested to supplement.\n7. please confirm whether the error value exceeds the average within the value “8.42 ± 9.34×10-2 mS/cm”. If it is wrong,\nplease correct it. Otherwise, please explain the reason for the excessive error.", "reviewer_3": "In this manuscript, the authors introduce SP as a novel descriptor to quantify the diffusional disorder of lithium ion pathways\nin SSEs, using a combination of the MSM, TPT and NNMD. The goal is to establish SP as a universal design principle for\nHTS of high-performance SSEs. The integration of information theory to analyze the complexity of ion diffusion pathways is\na potentially transformative approach, offering a thermodynamic perspective on transport dynamics. Here are some\ncomments:\n1. The electrochemical stability is the primary hurdle for practical SSEs. The work solely focuses on transport dynamics. Any\nproposed material must be stable under synthesis and operating conditions. The manuscript completely lacks analysis of\nstability of the predicted phases.\n2. The manuscript identifies Li4Cr2C4SO16 as a high-performance candidate, yet its calculated diffusion disorder is\nreported as low, similar to the poorly conducting reference phase. This contradiction must be resolved through deep\nmicrostructural and mechanistic analysis.\n3. The accuracy of MSM is sensitive to input parameters. The manuscript must prove the statistical robustness of its\nunderlying simulations.\n4. Provide a more in-depth microphysical explanation for the complementarity between structural configurational entropy and\npathway entropy"}, "author_response": "Response to Reviewers\n------------------------------------------------------------------------------------------------------------\nReviewer #1 (Remarks to the Author):\n------------------------------------------------------------------------------------------------------------\nDear authors,\nThank you for the insightful manuscript. It is very well written and indeed the newly defined\npath entropy is a central parameter to decouple merely the configurational entropy from\npotential enhancement in ionic diffusion in compositionally complex or high entropy\nmaterials.\nThe manuscript can be accepted for publication after addressing the following questions:\nRe: We sincerely appreciate your thoughtful appreciation of our work. We believe our\napproach using path entropy to design new solid-state electrolytes will be valuable for the\nbroader scientific community. Below, we have carefully addressed all questions you have\nraised, with original manuscript sentences underlined and revised text or new results\nhighlighted in red.\n1. Line 79: Li4Cr2C4SO16 is a computationally identified material. What is the main\nlimitation for its synthesis and experimental validation? Please clarify.\nRe: Thank you for your specific inquiry regarding the synthesis limitations of\nLi Cr C SO . The energy above hull (0.070 eV/atom) of Li Cr C SO is lower than\n4 2 4 16 4 2 4 16\nLi PS Cl (0.083 eV/atom), indicating better thermodynamic stability compared to\n6 5\nargyrodite systems. Regarding synthesis limitations, the decomposition products were\npredicted to be LiCr(CO ) and Li SO (Materials Project; [APL materials 1.1 (2013)]).\n3 2 2 4\nHence, potential synthesis strategies could involve these precursors. Notably, while Li SO\n2 4\nhas been experimentally synthesized, LiCr(CO ) (mp-770839) has not been validated\n3 2\naccording to the Materials Project database. However, similar structures like dolomite\n(CaMg(CO ) ) has already been experimentally verified [Sedimentology 62, 1749 (2015)],\n3 2\nwhich could inform precursor synthesis for LiCr(CO ) .\n3 2\n2. Line 149: Are these total LSs or per LCS?\nRe: These are total LSs, not per LCS. To clarify this, we have revised the sentence in the\nrevised manuscript to:\n“Specifically, a total of 10 LSs are identified for LPSCl-I, 24 LSs for LPSCl-II, 22 LSs for\nLPSCl-III, and 20 LSs for LSPSCl.” (line 151)\n3. Line 155: How many dynamic processes were identified for the other compositions? Is\nthis information included somewhere else?\nRe: Thank you for your thoughtful inquiry. We previously omitted similar plots for LPSCl-\nIII and LSPSCl due to space constraints. In the revised manuscript, we have now added\nback the implied timescale and C-K test plots for these systems. As shown in\nSupplementary Fig. 52 and 53, 10 dynamic processes were identified for one selected\nlithium ion in LPSCl-III and 4 dynamic processes for one selected lithium ion in LSPSCl.\nWe have also added the following cross-reference in the revised manuscript:\n“This test evaluates the consistency of transition probabilities between different discrete\nstates over time (Supplementary Fig. 51-53).” (line 458)\nSupplementary Figure 52. Chapman-Kolmogorov tests for one lithium ion in LPSCl-III.\na, Time evolution trajectory of a single lithium ion over 1.0 ns (20 fs time resolution)\nthrough various lithium states within LPSCl-III. b, Implied time scale plot of Markov\nprocess in LPSCl-III. c, Chapman-Kolmogorov test for the 11 most populated microstates.\nSupplementary Figure 53. Chapman-Kolmogorov tests for one lithium ion in LSPSCl. a,\nTime evolution trajectory of a single lithium ion over 1.5 ns (20 fs time resolution) through\nvarious lithium states within LSPSCl. b, Implied time scale plot of Markov process in\nLSPSCl. c, Chapman-Kolmogorov test for the 5 most populated microstates.\n4. Line 233: Please elaborate on the distinct rotation behaviour, if a higher barrier was\ncalculated. Other factors previously calculated like the inter-LCS diffusion seem to be more\nrelevant for enhancing ion conduction in LSPSCl than tetrahedra rotation? Please discuss\nin more detail.\nRe: Thank you for your question regarding the difference between rotation behavior and\ninter-LCS diffusion.\n(1) We have calculated the rotation barriers for all types of tetrahedral units in the LSPSCl\nsystem. These units fall into three distinct types:\na. The pristine [PS ]3- unit (shared across all LPSCl systems), with free energy profiles\n4\nplotted in Fig. 4e.\nb. The Si-substituted [SiS ]4- and [SiS Cl]3- (exclusive to LSPSCl), with free energy\n4 3\nsurfaces shown in Fig. 4f and Supplementary Fig. 40, respectively.\nWe have cross-referenced these results in the revised manuscript:\n“Notably, LSPSCl exhibits a slightly higher rotation barrier of pristine [PS ]3- tetrahedra\n4\nthan LPSCl-III (Fig. 4e and Supplementary Fig. 37), while a much higher barrier of ~ 300\nkJ/mol for Si-substituted tetrahedra ([SiS ]4- or [SiS Cl]3-) (Fig. 4f and Supplementary\n4 3\nFig. 38-40).” (Lines 230-233)\nFig. 4. Configurational disorder from anion framework. Free energy profiles for\ntetrahedral rotation in angular space at 300 K of the four argyrodite-type SSE phases: (d)\nLPSCl-III, the tilted [PS ]3- moiety (e) and [SiS ]4- moiety (f) in LSPSCl.\n4 4\nSupplementary Figure 40. Rotational free energy profile of the [SiS Cl]3- units in LSPSCl.\npolar and azimuthal angles, we conclude that the introduction of Si-centered tetrahedral\nunits raises unique rotation behavior directly linked to configurational disorder. In the\nrevised manuscript, we have added the following clarification:\n“Notably, LSPSCl exhibits a slightly higher rotation barrier of [PS ]3- tetrahedra than\nkJ/mol for Si-substituted tetrahedra ([SiS ]4- or [SiS Cl]3-) (Fig. 4f and Supplementary\n4 3\nFig. 38-40). While elevated barriers restrict bond rotation in [SiS ]4- and [SiS Cl]3-, the\n4 3\nintroduction of Si-substituted tetrahedral species generates distinct configurations of rigid\ntetrahedra units, which collectively enhance lithium-ion diffusion channel versatility.”\n(Lines 230-236)\n(2) Regarding the second question ( “Other factors previously calculated like the inter-LCS\ndiffusion seem to be more relevant for enhancing ion conduction in LSPSCl than tetrahedra\nrotation?”):\nYes, inter-LCS or intra-LCS diffusion would generally be more relevant for ion conduction\nin SSEs (not just LSPSCl) than tetrahedral rotation. As discussed in the section “Quantify\ndiffusional disorder through path entropy”, we focused on LPSCl-II and LPSCl-III\nbecause they share a similar LCS, enabling direct comparison. For LSPSCl, however, the\nnumber and type of LCS are fundamentally distinct from LPSCl-II, making the\nconstruction of analogous diffusion collective variables infeasible. Crucially, the primary\npurpose of calculating rotation barriers here is to provide readers with an intuitive\nfoundation for understanding how configurational entropy is built.\n5. Why do Sp and Se in the high-throughput screening for LSPSCl differ from previous\ncalculations shown in Fig.4?\nRe: Thank you for your thoughtful question regarding this value difference. This distinction\narises from different neural-network potential (NNP) (MACE, [Adv. Neural Inf. Process.\nSyst.35, 11423-11436 (2022)]) used in the high-throughput screening section.\nFor the high-throughput screening, we use a universal NNP, MACE-MPa potential (a\nPBE+U DFT-level foundation model; Supplementary Note 6.1), which is applicable\nacross nearly all materials. In contrast, for the other analyses (specific to our four argyrodite\nSSEs), we trained DeepMD-kit NNP using high-quality DFT data and active learning\n(Method section: Neural-network potential training).\nTo validate the robustness of MACE-MPa NNP for high-throughput screening, we have:\na. compared radial distribution functions (RDFs) with our datasets (Supplementary\nFig. 41), confirming consistency;\nb. verified that path entropy trends remain similar across both potentials in argyrodite\nsystems (Supplementary Fig. 42).\nCritically, for rational comparison in the high-throughput screening section, we explicitly\nreproduced LSPSCl (mp-1040451) and LPSCl-II (mp-985592) under the same MACE\nNNP, ensuring the analysis remains methodologically sound and directly comparable.\nHere are the details in Supplementary data explaining the MACE NNP we have used:\n“6.1. Validation of MACE neural-network potential.\nNeural network potential-based molecular dynamics simulations were performed utilizing\nthe MACE pretrained foundational model (medium size)3. The accuracy of this model was\nassessed by comparing radial distribution function (RDF) calculations to analogous\nresults from AIMD, as shown in (Supplementary Figure 41). While the MACE\nfoundational model’s accuracy is somewhat lower than that achieved with actively trained\nDeepMD-kit models (Supplementary Figure 11), it effectively captures the characteristic\nRDF peaks and demonstrates broad applicability across a wide range of materials, which\nis essential for high-throughput screening. Furthermore, consistent results for path entropy\nwere obtained using both MACE and DeepMD-kit (Supplementary Figure 42), providing\nstrong validation for the suitability of the MACE foundational model in this high-\nthroughput context.”\nSupplementary Figure 41. Radial distribution function plot of lithium with P, S, Li, and Cl\nelements at 300 K under NVT ensemble from MACE potential3 for a, LPSCl-II, b, LSPSCl,\nand c, LPSCl-III. The dotted and solid lines represent the calculation from DFT and MACE\npotential, respectively.\nSupplementary Figure 42. Path entropy comparison of NNMD results from DeepMD and\nMACE potentials.\n6. Line 461: Why is the standard error in the same order as the probability? Or is an error\nof 0.01 acceptable in this case? Please clarify.\nRe: Thank you for your thoughtful question and for catching this important clarification\npoint. The range [0.14, 0.16] refers to the probability values (not standard errors) for which\nionic transfer is considered effective. Specifically, we define effective ionic transfer as\noccurring when the probability exceeds 0.15. We have updated the manuscript accordingly\nto ensure no ambiguity remains:\n“Ionic transfer between different states and is considered effective if the probability\nexceeds 0.15. The standard error of path entropy is calculated within the transfer\nprobability range of [0.14, 0.16].” (Lines𝑖 482-4𝑗84)\n7. How does the model consider the influence of the electric field? Experimentally, long\nrange ionic conductivity is mostly determined by applying a bias (AC). Is it, nevertheless,\npossible to correlate the simulated and experimental findings? Please clarify in a short\ndiscussion.\nRe: Thank you for your insightful question. We appreciate this opportunity to clarify the\nscope of our model. The assumption that most periodic units remain stable is indeed\nfoundational to our approach; this is consistent with the framework used in our\nmetadynamics simulations, where external bias potentials were applied to lithium atoms to\nexplore free energy surfaces.\nIn experimental context, the AC electric field (e.g., 50 Hz) reverses direction 100 times per\nsecond (50 cycles per second). This time-dependent behavior can be directly implemented\nin simulations using tools like “fix efield” command in LAMMPS, which adds time-\nvarying external potentials. Crucially, by integrating neural network potentials, we can also\nachieve realistic simulation of thousands of atoms within seconds. This aligns with the\nneed for efficient exploration of electric field effects in complex systems.\nWe believe this approach is highly applicable to real-world scenarios where periodic\nstability dominates, and we are eager to apply it in future work to bridge computational\nfindings with experimental observations.\nWe would like to thank Reviewer #1 again for the useful suggestions which are important\nfor the improvement of our work.\n------------------------------------------------------------------------------------------------------------\nReviewer #2 (Remarks to the Author):\n------------------------------------------------------------------------------------------------------------\nIn this paper, the authors proposed a descriptor quantifying diffusion pathway diversity—\npath entropy (Sp), addressing the limitation of traditional configurational entropy that\nfocuses on structural disorder. Through combining Markov state models (MSMs) with\ntransition path theory (TPT), this work revealed the correlation between local\nenvironments and ionic diffusion in solid-state electrolytes. Furthermore, this work\ndeveloped a path entropy-based high-throughput screening process for inorganic\nthiophosphates, enabling efficient identification of high-performance candidate materials.\nIn conclusion, I recommend that this work can be considered to publish in Nature\nCommunications after Major revision. Several comments and questions are included as\nfollow:\nRe: We sincerely thank Reviewer #2 for his/her thorough review of our work and for\nhighlighting how our approach efficiently identifies high-performance candidates. We\nappreciate her/his recommendation for acceptance after major revisions that could\nsignificantly improve the generality and robustness of our calculations. Below, we have\ncarefully addressed all questions Reviewer #2 has raised, with original manuscript\nsentences underlined and revised text or new results highlighted in red.\n1. Why did this work focus merely on verifying the strategies of lithium vacancy creation\nand anion substitution, instead of exploring other entropy-driven approaches (e.g., cation\ndoping or multicomponent integration)? By the way, investigating these additional\nmethods could further demonstrate the general applicability of pathway entropy as a\ndesign criterion.\nRe: We sincerely appreciate your insightful comments regarding exploring other entropy-\ndriven strategies. We fully agree that incorporating additional entropy-driven\nmethodologies would further strengthen the robustness and general applicability of our\nframework.\nWe selected the strategies of lithium vacancy creation and anion substitution for two key\nreasons: first, they provide a straightforward approach to elucidating the concept of path\nentropy; second, these methods have been well measured and recorded through\nexperimental studies. To this end, our methodology specifically highlights two\nexperimentally validated strategies for entropy-driven design: (1) engineering of the\nflexible lithium coordination shell (LCS) through vacancy creation or cation doping, and\n(2) engineering the rigid anion framework (such as [PS ]3-) through substitution or\n4\ndoping.\nImportantly, the two approaches have been experimentally demonstrated to significantly\nenhance superionic conductivity, which arise many attentions. [Nat. Energy. 1, 16030\n(2016); Angew. Chem. Int. Ed. 58, 8681-8686 (2019)] In contrast, many alternative\nstrategies yield limited improvements [Energy Environ. Mater. 7, e12729 (2024); Energy\nStorage Mater. 45, 484-493 (2022)] compared to these two approaches. By focusing on\nthese two well-established, high-impact strategies (substitution and vacancy engineering),\nwe have constructed representative argyrodite systems (LPSCl-III and LSPSCl) that\ndemonstrate the practical applicability of path entropy in real materials.\nWe also strongly align with your suggestion to extend validation through additional\nmethodologies. However, for each strategy (e.g., lithium vacancy introduction, substitution,\nmulticomponent integration …), there are nearly infinite optimization pathways within\nsystems like LPSCl, LGPS, and others. [Joule6, 543-587 (2022)] Therefore, demonstrating\nthese two foundational entropy-driven aspects with solid experimental evidence provides\na highly instructive, scalable foundation for future work. Nevertheless, to further validate\nthe general applicability of path entropy, we have implemented a high-throughput\nscreening approach targeting the broader sulfide compound space.\n2. The authors defined escape entropy (Se) as “the contribution from transitions from the\noriginal LCS to external states”, but failed to clarify the standard for defining the “original\nLCS” (i.e., how to define the boundaries of LCS). We suppose that relevant numerical\nstandards can be added.\nRe: Thank you for your careful examination and insightful comment regarding the\nfundamental role of LCS boundaries in our approach. The boundaries are determined\nthrough a Voronoi partition process, which we have provided details in Supplementary\nNote 3. In Supplementary Note 3, we have provided details of the method we used in\ndetermining the number of LCS and the partitioned space for each LCS. We have also\nprovided numerical standards for a different cutoff range of LCS boundaries from 1.50 to\n2.00 Å. The numerical values and standard errors are also provided in Supplementary\nTables 6 and 7.\nTo make it clearer, we now have revised related parts in the revised manuscript with a short\nexplanation of methods provided in Supplementary Note 3:\n“We then discretize each LCS into partitioned 3D subspaces representing discrete lithium\nstates (LSs) using classical Voronoi partitioning (Fig. 2i, details in partitioned boundaries\nand error estimation are provided in Supplementary Note 3)” (Lines 149-150)\nHere’s the corresponding Supplementary Note 3:\n“3. Supplementary Note 3\nPeriodic K-means and Voronoi partition.\nTo identify local coordination shells (LCSs), we implemented periodic K-means clustering\nalgorithm using the scikit-learn package9. The number of LCSs, denoted as K (where K is\nless than or equal to the total number of lithium atoms in the unit cell), was determined by\nminimizing the within-cluster sum of squares (WCSS). Following cluster identification, a\nVoronoi partition including application of periodic boundary conditions was performed in\nthree dimensions for each LCS10. For Voronoi partitioning calculations, we defined a 3D\nbox extending 1.75 Å from the maximum/minimum atomic positions of each LCS. Lithium\natoms are considered to transition from an original LCS to a new LCS only when they move\nfrom the overlapping region of the original LCS to another LCS. Uncertainty analysis\nacross this cutoff range (1.50–2.00 Å) was systematically evaluated and summarized in\nSupplementary Tables 6and 7.”\n3. Based on the last review, the gap of LCS among LPSCl-I, II, III, and LSPSCl (3/4/4/5\ntypes) and whether it is related to the intrinsic structure of the materials are expected to\nbe explained.\nRe: Thank you for highlighting this important aspect of our approach. It’s related to the\nintrinsic structure of materials, especially the distribution of lithium ions. To ensure our\nmethod is well generalized to other materials, we use a periodic K-means algorithm to\ndetermine the number of LCS. We have provided the details of this method in\nSupplementary Note 3. To ensure clarity and accessibility, we have also made\nclarifications in the revised manuscript:\n“Following a periodic K-means clustering method (Supplementary Note 3), which\ndetermines the optimal number of LCSs, lithium ions are assigned to LCSs characterized\nby distinct spatial and angular distributions.” (Lines 144-145)\n4. In the flowchart that depicting the multistep high-throughput screening for identifying\npromising SSE candidates (Fig.5a), the way to determine the threshold setting of “MSD >\n0.018 nm2” in Step 3 remains vague. Similarly, in Step 4, why “200 J/mol/K” was chosen\nas the critical value? Furthermore, is this value universal or specific for sulfide SSE\nsystems?\nRe: Thank you for your insightful comment regarding our screening criteria. We clarify the\nrationale for each as follows:\n(1) For the first MSD criterion, we selected this value (0.018 nm2) based on calibration\nwith the argyrodite system Li PS Cl (1.92 × 10-2 nm2; Supplementary Table 9). This\n6 5\nslightly lower value ensures a margin of ~20 SSE candidates for baseline ionic diffusion\nperformance. It is not a universal value but rather context-dependent: For initial\nscreening with short-time MSD calculations involving tens of thousands of candidates,\na higher threshold may be appropriate.\n(2) For the path entropy criterion, the 200 J/mol/K was chosen based on calibration with\ntwo systems: LPSCl-type (Li PS Cl; 273.52 ± 0.00 J/mol/K), and LGPS-type\n6 5\n(Li Si(PS ) ; 167.51 ± 16.24 J/mol/K). This value is transferable across other systems\n10 6 2\nwhen using the same machine learning potential (e.g., MACE). However, it is not\nuniversally fixed, it may be adjusted for looser or tighter selection criteria depending\non high-throughput screening procedure.\n5. It is necessary for authors to give an in-depth analysis of the fundamental relationship\namong Sp, Se and Sc. For example, several systems (e.g., LSPSCl) have both high\nconfigurational entropy and high path entropy, while others (e.g., LPSCl-III) show low\nconfigurational entropy but extremely high path entropy. Is this discrepancy related to the\nchanges in lithium coordination environment?\nRe: Thank you for your thoughtful question regarding the relationship among different\ntypes of entropies. The system-level disorder in SSE originates from diffusional disorder\n(quantified by S ) and configurational disorder (quantified by S ). Escape entropy S is part\np c e\nof S , representing entropic contribution of long-range diffusion.\np\nIn LSPSCl, substituting Si with P directly introduces configurational disorder into the host\nframework, resulting in high S . For LPSCl-III, lithium vacancies are localized within the\nc\nLCSs rather than altering the host framework structure. Therefore, the configurational\nentropy change, relative to pristine LPSCl-II, is minimal.\nBoth systems exhibit high path entropy because these entropy-driven mechanisms, whether\nthrough substitution (LSPSCl) or vacancy introduction (LPSCl-III), effectively influence\nlithium-ion conduction within the LCSs. This conduction behavior is directly quantified by\npath entropy. Therefore, this discrepancy is related to the changes in LCS, whether the LCS\nhas been directly engineered or not.\nWe have now added this detailed summary in the revised manuscript under section\nConnection between configurational entropy and path entropy:\n“Entropy contributions related to ionic conduction in SSEs arise from both the flexible\nLCSs containing mobile lithium ions and the rigid host framework that forms the structural\nbackbone. Disorder in these systems originates from two components: diffusional disorder\nassociated with lithium-ion motion and configurational disorder in the host lattice.\nConsequently, total entropy is partitioned into path entropy from the flexible LCSs and\nconfigurational entropy from the rigid host framework. By combining these two\ncomponents, the system-level disorder can be quantified 𝑆r 𝑝 ather than assessed solely\nthrough the disorder in𝑆d 𝑐 uced by the lithium-ion conduction or host framework\nconfigurations.\nIt is noteworthy that while systems with high may exhibit high , no direct causal\nrelationship exists between these quantities. Instead, they represent complementary facets\nof solid-state ionic conductors. Joint analys𝑆i 𝑝 s of and en𝑆a 𝑐 bles unambiguous\nidentification of entropy gains in entropy-driven-designed systems. Compared to minor\nvariations in the by different entropy-driven strategi𝑆e 𝑝 s (e.g., v𝑆a 𝑐 cancy introduction), the\n, a metric quantifying diffusional disorder, aligns strongly with ionic diffusion\nperformance. Its 𝑆v 𝑐 alues range from 0.0 J/mol/K (LPSCl-I) to 415.72 ± 6.02 J/mol/K\n𝑆(L 𝑝 PSCl-II), 553.91 ± 54.42 J/mol/K (LSPSCl), and 2598.16 ± 112.58 J/mol/K (LPSCl-III)\n(Fig. 4g). These results confirm that the provides a more direct and quantitatively robust\nmetric for assessing ionic diffusion than the . A more nuanced separation of escape\nentropy from path entropy , enable𝑆s 𝑝 the clear identification of long-range diffusion in\nLSPSCl (91.70 ± 15.82 J/mol/K) and LPSCl-III𝑆 𝑐 (513.44 ± 26.86 J/mol/K).”\n𝑆𝑒 𝑆𝑝\n6. For the novel candidate material Li4Cr2C4SO16, only ionic conductivity at 300 K is\nprovided. To determine its activation energy and high-temperature stability, the\ntemperature dependant EIS (e.g., Arrhenius plot) is suggested to supplement.\nRe: Thank you for this constructive comment. As previously noted, the diffusion\ncoefficients plot for Li Cr SO was provided in the inset of Fig. 5b. Following your\n4 2 16\nsuggestion, we have now explicitly added the Arrhenius plot for this material in\nSupplementary Fig. 44, with a calculated activation energy of 0.18 eV. This\nsupplementary figure is now explicitly referenced in the revised manuscript:\nIn the revised manuscript, the supplementary Arrhenius plot is referenced:\n“Li Cr C SO achieves ionic conductivity of 5.05± 0.23mS/cm (Supplementary Fig. 44;\n4 2 4 16\nactivation energy 0.18 eV).” (Line 318)\nSupplementary Figure 44. Arrhenius plot of ionic conductivities of Li Cr C SO . The\n4 2 4 16\ncorresponding activation energy is 0.18 eV.\n7. please confirm whether the error value exceeds the average within the value “8.42 ±\n9.34×10-2 mS/cm”. If it is wrong, please correct it. Otherwise, please explain the reason\nfor the excessive error.\nRe: Thank you for this careful examination and identifying this misleading expression. The\naccurate value is 8.42 ± (9.34 × 10-2) mS/cm (not (8.42 ± 9.34) ×10-2 mS/cm). We have\nnow corrected this in the revised manuscript:\n“… and 8.42 ± 0.093 mS/cm (close to reported value9 of 9.4 mS/cm), respectively.” (Line\n112)\n“Notably, this conductivity rivals that of LPSCl-III (8.42 ± 0.093 mS/cm)” (Line 319)\nWe would like to thank Reviewer #2 again for the useful suggestions which are important\nfor the improvement of our work.\n------------------------------------------------------------------------------------------------------------\nReviewer #3 (Remarks to the Author):\n------------------------------------------------------------------------------------------------------------\nIn this manuscript, the authors introduce SP as a novel descriptor to quantify the\ndiffusional disorder of lithium ion pathways in SSEs, using a combination of the MSM, TPT\nand NNMD. The goal is to establish SP as a universal design principle for HTS of high-\nperformance SSEs. The integration of information theory to analyze the complexity of ion\ndiffusion pathways is a potentially transformative approach, offering a thermodynamic\nperspective on transport dynamics. Here are some comments:\nReply: We sincerely thank Reviewer #3 for his/her detailed review and insightful\ncomments on our approach, which is potentially transformative for analyzing transport\ndynamics. Below, we have carefully addressed all questions Reviewer #3 has raised, with\noriginal manuscript sentences underlined and revised text or new results highlighted in red.\n1. The electrochemical stability is the primary hurdle for practical SSEs. The work solely\nfocuses on transport dynamics. Any proposed material must be stable under synthesis and\noperating conditions. The manuscript completely lacks analysis of stability of the predicted\nphases.\nRe: Thank you for pointing out the very fundamental property of SSE should be considered\nduring the high-throughput screening. We fully agree that electrochemical stability\nrepresents the primary hurdle for practical SSEs. Nevertheless, we employ the energy\nabove hull ( ) which is well accepted to be the essential metric for assessing\nelectrochemical stability, hence directly evaluating the stability of all screened SSEs. We\nhave listed th𝐸e h ull for all candidates meeting the criterion < 0.085 eV/atom\n(comparable to Li PS Cl system of 0.083 eV/atom) (Supplementary Table 9). Since this\n6 5\nstability criterion𝐸 h w ul a l s implemented in Step 2 of our high-thr𝐸o h u u g ll hput screening, all\npredicted candidates satisfy < 0.085 eV/atom, confirming their computational\nstability.\n𝐸hull\nTo make it clearer, in the revised manuscript, the procedure we’ve implemented in our\nmanuscript has been explicitly described:\n“Subsequent filtering criteria included: (i) a band gap greater than 2.0 eV ( )\nand (ii) an energy above hull ( ) 85 meV per atom (comparable to LPSCl-II). This\nreduced the candidate pool to 509 materials. While these candidates sat𝐸is 𝑏 fi 𝑎 e 𝑛 d 𝑑 t≥he 2i.n0i t𝑒ia𝑉l\nscreening criteria for structura𝑬l 𝒉 a 𝒖 n 𝒍𝒍 d e≤lectronic stability, ionic mobility remains a critical\nfactor for SSE performance.”\n2. The manuscript identifies Li4Cr2C4SO16 as a high-performance candidate, yet its\ncalculated diffusion disorder is reported as low, similar to the poorly conducting reference\nphase. This contradiction must be resolved through deep microstructural and mechanistic\nanalysis.\nRe: Thank you for your careful review and this insightful comment. Following your\nsuggestion, we have now resolved it with additional description and mechanistic analysis\nof this new compound.\nThe calculated path entropy of Li Cr C SO is 263.98 ± 10.42 J/mol/K, comparable to the\n4 2 4 16\nLi PS Cl (LPSCl-II) at 273.52 ± 0.00 J/mol/K. Crucially, the ionic conductivity of the\n6 5\nnovel Li Cr C SO (5.05± 0.23 mS/cm)is significantly higher than that of LPSCl-II (1.53\n4 2 4 16\n× 10-3 ± 1.40 × 10-4 mS/cm).\nThis increased ionic conductivity arises from a greater number of lithium conduction states\nwith a wider distribution relative to LPSCl-II. Specifically, Li Cr C SO exhibits 5\n4 2 4 16\nconnected LCSs, enabling efficient transfer between states (Supplementary Fig. 45). As\nshown in Supplementary Fig. 46-50, comparing with the kinetics of lithium conduction\nin LPSCl-II or LPSCl-III, the mean first passage time (MFPT) profile of Li Cr C SO lies\n4 2 4 16\nbetween LPSCl-II and LPSCl-III. This intermediate behavior occurs because the connected\nLCSs permit unforbidden out-LCS conduction (unlike LPSCl-II; Supplementary Fig. 14-\n17) but without the intensity of conduction in LPSCl-III (Supplementary Fig. 18-21).\nConsequently, the ionic conduction in this material is less constrained than in LPSCl-II.\nThese results are now fully incorporated into Supplementary Fig. 45-50. A detailed\nmicrostructural and mechanistic analysis has also been added to the revised manuscript:\n“Notably, this conductivity rivals that of LPSCl-III (8.42 ± 0.093 mS/cm), where long-range\nionic diffusion is activated via lithium-ion vacancies and site disorder. Li Cr C SO\n4 2 4 16\nexhibits 32 lithium sites with sparse distribution compared to LPSCl-II, resulting in 5\nconnected LCSs (Supplementary Fig. 45). This LCS configuration enables enhanced\nlithium-ion transport kinetics, which fall between the constrained kinetics of LPSCl-II\n(Supplementary Fig. 14-17) and the fully activated diffusion in LPSCl-III (Supplementary\nFig. 18-21). The partially unlocked out-LCS conduction facilitates higher ionic\nconductivity (Supplementary Fig. 44) despite comparable path entropy to LPSCl-II.\nThe nearly-zero S (1.8 J./mol/K; Supplementary Table 9) in Li Cr C SO confirms\ne 4 2 4 16\npredominantly short-range diffusion mechanisms, analogous to pristine LPSCl-II.\nThis characteristic suggests that employing entropy-driven strategies (e.g., vacancy\nengineering or site disorder) shown in step 6 of Fig. 5a could further significantly enhance\nits superionic performance.” (Lines 319-332)\nSupplementary Figure 45. Distribution of LCSs and path entropies in Li Cr C SO .\n4 2 4 16\na, distribution of LCSs in Li Cr C SO . b, bar plot of LCS-based path entropy and escape\n4 2 4 16\nentropy.\nSupplementary Figure 46. MFPT profile of lithium hopping in Li Cr C SO at 300 K\n4 2 4 16\nof LCS-1. The initial lithium states in LCS-1 are denoted with a cyan rectangle. Blocks\ncolored blue represent no transition happens or have MFPT value greater than 10000 ps.\nSupplementary Figure 47. MFPT profile of lithium hopping in Li Cr C SO at 300 K\n4 2 4 16\nof LCS-2. The initial lithium states in LCS-2 are denoted with a cyan rectangle. Blocks\ncolored blue represent no transition happens or have MFPT value greater than 10000 ps.\nSupplementary Figure 48. MFPT profile of lithium hopping in Li Cr C SO at 300 K\n4 2 4 16\nof LCS-3. The initial lithium states in LCS-3 are denoted with a cyan rectangle. Blocks\ncolored blue represent no transition happens or have MFPT value greater than 10000 ps.\nSupplementary Figure 49. MFPT profile of lithium hopping in Li Cr C SO at 300 K\n4 2 4 16\nof LCS-4. The initial lithium states in LCS-4 are denoted with a cyan rectangle. Blocks\ncolored blue represent no transition happens or have MFPT value greater than 10000 ps.\nSupplementary Figure 50. MFPT profile of lithium hopping in Li Cr C SO at 300 K\n4 2 4 16\nof LCS-5. The initial lithium states in LCS-5 are denoted with a cyan rectangle. Blocks\ncolored blue represent no transition happens or have MFPT value greater than 10000 ps.\n3. The accuracy of MSM is sensitive to input parameters. The manuscript must prove the\nstatistical robustness of its underlying simulations.\nRe: We fully agree with your point that robustness and reproducibility form the foundation\nof a new approach. To ensure the robustness of our methodology, particularly the MSM\nconstruction and associated Voronoi partition analysis, we conducted three independent\nsimulations, each with a minimum trajectory length of 1.0 ns, as input for MSM modeling.\nThe final path entropy and escape entropy values are derived from aggregating results\nacross all three simulations.\n(1) The statistical validation of this robust approach is detailed in the Method section\nCalculation of lithium flux and path entropy:\n“… For four types of SSE, three independent simulations were performed to compute\nprobability densities and associated path entropies. The mean path entropies fall within a\n99.5% confidence interval.” (Lines 484-486)\n(2) For parameters affecting MSM modeling (e.g., Voronoi partition boundaries), we report\nstandard errors by counting the numerical values of partition boundaries (Supplementary\nNote 3):\n“Periodic K-means and Voronoi partition.\nTo identify local coordination shells (LCSs), we implemented periodic K-means clustering\nalgorithm using the scikit-learn package9. The number of LCSs, denoted as K (where K is\nless than or equal to the total number of lithium atoms in the unit cell), was determined by\nminimizing the within-cluster sum of squares (WCSS). Following cluster identification, a\nVoronoi partition including application of periodic boundary conditions was performed in\nthree dimensions for each LCS10. For Voronoi partitioning calculations, we defined a 3D\nbox extending 1.75 Å from the maximum/minimum atomic positions of each LCS. Lithium\natoms are considered to transition from an original LCS to a new LCS only when they move\nfrom the overlapping region of the original LCS to another LCS. Uncertainty analysis\nacross this cutoff range (1.50–2.00 Å) was systematically evaluated and summarized in\nSupplementary Tables 6and 7.”\nThese measures confirm the reproducibility and reliability of our analysis framework.\n4. Provide a more in-depth microphysical explanation for the complementarity between\nstructural configurational entropy and pathway entropy.\nRe: Thank you for your insightful suggestion to provide a detailed explanation of path\nentropy and configurational entropy. As we illustrated in Fig. 1, SSEs can be decomposed\ninto two distinct components: (1) A flexible local coordination shell (LCS) containing\nmobile lithium ions, and (2) A rigid host framework. This decomposition allows us to\nseparate total disorder into two physically meaningful contributions:\n(1) Configurational entropy S quantifies disorder arising from the rigid host framework.\nc\n(2) Path Entropy S quantifies disorder associated with the flexible LCS, representing the\np\ndiffusional disorder of lithium ions.\nBy explicitly distinguishing these components, we can now assess total disorder in ionic\nsystems, not merely lithium conduction effects or host framework distortions.\nWe have now incorporated this in-depth explanation in the revised manuscript under\nsection Connection between configurational entropy and path entropy:\n“Entropy contributions related to ionic conduction in SSEs arise from both the flexible\nLCSs containing mobile lithium ions and the rigid host framework that forms the structural\nbackbone. Disorder in these systems originates from two components: diffusional disorder\nassociated with lithium-ion motion and configurational disorder in the host lattice.\nConsequently, total entropy is partitioned into path entropy from the flexible LCSs and\nconfigurational entropy from the rigid host framework. By combining these two\ncomponents, the system-level disorder can be quantified 𝑆r 𝑝 ather than assessed solely\nthrough the disorder induced by the lithium-ion conduction or host framework\nconfigurations.\nIt is noteworthy that while systems with high may exhibit high , no direct causal\nrelationship exists between these quantities. Instead, they represent complementary facets\nof solid-state ionic conductors. Joint analys𝑆i 𝑝 s of and en𝑆a 𝑐 bles unambiguous\nidentification of entropy gains in entropy-driven-designed systems. Compared to minor\nvariations in the by different entropy-driven strategi𝑆e 𝑝 s (e.g., v𝑆a 𝑐 cancy introduction), the\n, a metric quantifying diffusional disorder, aligns strongly with ionic diffusion\nperformance. Its 𝑆v 𝑐 alues range from 0.0 J/mol/K (LPSCl-I) to 415.72 ± 6.02 J/mol/K\n𝑆(L 𝑝 PSCl-II), 553.91 ± 54.42 J/mol/K (LSPSCl), and 2598.16 ± 112.58 J/mol/K (LPSCl-III)\n(Fig. 4g). These results confirm that the provides a more direct and quantitatively robust\nmetric for assessing ionic diffusion than the . A more nuanced separation of escape\nentropy from path entropy , enable𝑆s 𝑝 the clear identification of long-range diffusion in\nLSPSCl (91.70 ± 15.82 J/mol/K) and LPSCl-III𝑆 𝑐 (513.44 ± 26.86 J/mol/K).”\n𝑆𝑒 𝑆𝑝\nWe would like to thank Reviewer #3 again for the useful suggestions which are important\nfor the improvement of our work."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Dear Authors,\nthank you for thoroughly addressing all comments and making all necessary changes to the manuscript and supporting\ninformation.\nI suggest accepting the manuscript for publication.", "reviewer_2": "The authors have addressed the comments well. I recommend that this manuscript could be published in Nature\nCommunications.", "reviewer_3": "The authors have addressed several of my previous concerns. However, before the manuscript can be recommended for\npublication, there are two critical technical points regarding the simulation settings and data reliability that must be clarified:\n1. In the SI, the authors stated that the energy cutoff for the projector augmented wave method was set to 450 eV. However,\nfor systems containing Li, the standard PAW potentials typically recommend a maximum cutoff of approximately 499.034 eV\nfor high-precision calculations. Could the authors justify the use of 450 eV, which is lower than the recommended value for\nLi? Please provide evidence to demonstrate that this choice does not compromise the accuracy.\n2. On lines 302-303, the author stated “Candidates with measurable lithium-ion mobility, defined as MSDs > 0.018 nm2 over\n6.0 ps of NNMD simulation”. A 6.0 ps window is relatively short for extracting diffusion properties in complex solid-state\nelectrolytes. More importantly, the authors must ensure that the system has reached both thermodynamic and kinetic\nequilibrium before the commencement of the MSD data collection."}, "author_response": "Manuscript ID: NCOMMS-25-93405A\nTitle: \"Path Entropy-driven Design of Solid-State Electrolytes\"\nAuthor(s): Qiye Guan, Kaiyang Wang, Jingjie Yeo, Yongqing Cai\nFebruary 18, 2026\nResponse to Reviewers\n------------------------------------------------------------------------------------------------------------\nReviewer #1 (Remarks to the Author):\n------------------------------------------------------------------------------------------------------------\nDear Authors,\nthank you for thoroughly addressing all comments and making all necessary changes to\nthe manuscript and supporting information. I suggest accepting the manuscript for\npublication.\nReply: We sincerely appreciate your recommendation for acceptance of our revised\nmanuscript.\n------------------------------------------------------------------------------------------------------------\nReviewer #2 (Remarks to the Author):\n------------------------------------------------------------------------------------------------------------\nThe authors have addressed the comments well. I recommend that this manuscript could\nbe published in Nature Communications.\nReply: We sincerely appreciate your recommendation for acceptance of our revised\nmanuscript.\n------------------------------------------------------------------------------------------------------------\nReviewer #3 (Remarks on code availability):\n------------------------------------------------------------------------------------------------------------\nThe authors have addressed several of my previous concerns. However, before the\nmanuscript can be recommended for publication, there are two critical technical points\nregarding the simulation settings and data reliability that must be clarified:\nReply: We sincerely appreciate Reviewer #3 for his/her detailed review and thorough\nexamination of our simulations. Below, we have carefully addressed all questions\nReviewer #3 has raised, with original manuscript sentences underlined and revised text or\nnew results highlighted in red.\n1\n1. In the SI, the authors stated that the energy cutoff for the projector augmented wave\nmethod was set to 450 eV. However, for systems containing Li, the standard PAW potentials\ntypically recommend a maximum cutoff of approximately 499.034 eV for high-precision\ncalculations. Could the authors justify the use of 450 eV, which is lower than the\nrecommended value for Li? Please provide evidence to demonstrate that this choice does\nnot compromise the accuracy.\nReply: Thank you for this careful observation. We fully agree that simulations must be\ncomputed with maximal accuracy when possible. We would like to clarify that there exist\ntwo types of pseudopotentials generated for Li depending on the selection of valence\nelectrons (2s1 or1s2 2s1). We adopted the standard Li PAW potential with 2s1 (recommend\nenergy max (ENMAX) = 140 eV) in our calculations. Our energy cutoff of 450 eV is\napproximately 3.2×ENMAX for the standard Li potential (ENMAX = 140 eV) and exceeds\nthe ENMAX values of all elements in the system, ensuring well-converged results for\nstructural stability. Regarding the slightly higher cutoff value of 499.034 eV mentioned by\nthe Reviewer, it corresponds to the Li_sv (1s2 2s1) potential, which treats the 1s electrons\nas valence states.\nThis calculation setting (450 eV cutoff with standard Li potential) was selected for\nidentifying stable LPSCl-III structures via cluster-expansion (CE), as detailed in the\nMethods section. For CE model preparation, we screened 340 structures generated through\nenumeration. Given our focus on structural properties rather than formation energies or\nelectronic structures, the valence 2s electron is most relevant and standard Li potential is\ncomputationally efficient for this large dataset. This approach avoids unnecessary cost\nwhile maintaining sufficient accuracy for relative energy comparisons.\nNevertheless, to justify this choice, as inspired by the Reviewer, we had also tested 10\nrandomly selected structures simulated and compared with both standard Li (energy cutoff\n= 450 eV) and Li_sv (energy cutoff = 500 eV) pseudopotentials. As shown in Figure R1,\nboth settings consistently distinguish stable structures, with deviations mostly limited to\nabsolute energy values. Thus, the standard Li PAW potential (ENMAX= 140 eV) at 450\neV energy cutoff provides a cost-effective and reliable basis for our CE model initialization.\nunder different cutoffs for identifying stable LPSCl-III structures.\n2. On lines 302-303, the author stated “Candidates with measurable lithium-ion mobility,\ndefined as MSDs > 0.018 nm2 over 6.0 ps of NNMD simulation”. A 6.0 ps window is\nrelatively short for extracting diffusion properties in complex solid-state electrolytes. More\nimportantly, the authors must ensure that the system has reached both thermodynamic and\nkinetic equilibrium before the commencement of the MSD data collection.\nReply: We sincerely appreciate your insightful comment regarding the MSD calculation.\nAs suggested by the Reviewer, we conduct more examination to evaluate the issue of time\nwindow to derive the MSD. We indeed found that a 6.0 ps window is relatively insufficient\nfor high-throughput screening and therefore have implemented the following\nimprovements:\n(1) Extended simulation window: We conducted a 12.0 ps NNMD simulation for all\nthe candidates with the first 2.0 ps discarded to ensure full thermal relaxation at\nroom temperature prior to analysis.\n(2) Revised screening criterion: To prevent omission of potential superionic candidates,\nwe lowered the MSD threshold from 0.018 nm2 to 0.010 nm2 in the third screening\nround. This adjustment accounts for systematic overestimation of MSD due to the\npreviously relatively short simulation window and incomplete system relaxation.\ncompounds (SeeSupplementary Table 9), in which 17 compounds were retained from the\nprevious list and 10 compounds are newly added highlighted in red. Subsequent path\nentropies (S ) calculations of these 10 new candidates identified three experimentally\np\nconfirmed high-performance superionic conductors with S > 200 J/mol/K: (1) Argyrodite-\np\ntype: Li PS Br and Li PS [ACS Appl. Mater. Interfaces.11, 6015–6021 (2019); Chem. Eur.\n6 5 7 6\nJ. 16, 5138–5147 (2010)] and (2) Thioborate-type [J. Mater. Res. 37, 3269–3282 (2022)]\nLi B S [Solid State Ion.78, 305–313 (1995)]. Additionally, we also identified four other\n5 7 13\nexperimentally or theoretically validated candidates with lower S values: Li TiS [J. Am.\np 4 4\nChem. Soc.139, 8796–8799 (2017)], Li CuS [ACS Appl. Energy Mater.4, 20–24 (2020)],\n3 2\nLi GeS [Solid State Ion.154-155, 789–794 (2002)], and Li TiS [Npj Comput. Mater.11,\n4 4 8 6\n67 (2025)].These results are now fully incorporated into Fig. 5 andSupplementary Table\n9.\nWe have revised the manuscript accordingly regarding the updated results:\n“Candidates with measurable lithium-ion mobility, defined as MSDs > 0.010 nm2 over 12.0\nps of NNMD simulation, were retained. This procedure narrowed the list to 27 structurally\nstable compounds with detectable ionic motion (Supplementary Table 9).” (Lines 303-305)\n“…, with the S > 200.0 J/mol/K, including experimentally verified SSEs such as\np\nargyrodite-type compounds14,29,30 (LPSCl-II (Li PS Cl), Si-substituted LSPSCl13,14\n6 5\n(Li Si P S Cl ), Li PS Br, and Li PS 40), thioborate-type41 Li B S 42,Li P S 43,rock-salt\n20 3 3 23 1 6 5 7 6 5 7 13 7 3 11\nsulfide Li NbS 44, and LGPS-type structures45 (Li Ge(PS ) , Li Sn(PS ) ).” (Lines 309-\n3 4 10 6 2 10 6 2\n313)\nThe high-throughput screening procedure described in Supplementary Note 6.2 has also\nbeen revised:\n“…, NNMD simulations were conducted over a total time of 12.0 ps using a 2×2×2\nsupercell. To ensure the system reaches full equilibrium at room temperature, the first 2.0\nps of simulated trajectories were discarded before MSD analysis. To generate path\nentropies for the final production (4th screening) run encompassing all 27 candidate\nmaterials, NNMD simulations were performed for at least 800.0 ps using larger supercells\n(typically a 3×3×3 supercell, varying depending on cell size).” (Lines 389-394)\n4\nentropy. a, Flowchart depicting the multistep high-throughput screening procedure for\nidentifying promising SSE candidates. b, Summary of the path entropy and the escape\nentropy screening results (top 20 out of 27 candidates) from the step 4 of the workflow.\nCandidates with favorable lithium-ion diffusion characteristics are highlighted with cyan\nrectangles. The inset shows diffusion coefficient D as a function of the reciprocal\ntemperature for Li Cr C SO .\n4 2 4 16\nSupplementary Table 9. Summary of 27 screened superionic candidates.\nEnergy\nabove\nBand MSD at Path Escape\nSpace hull Chemical\nNo. Identifier gap 300K entropy entropy\nGroup per formula\n(eV) (nm2) (J/mol/K) (J/mol/K)\natom\n(eV)\nmp- 1016.57 ± 254.69 ±\n1 6 2.39 0.035 Li Si P S Cl 0.0118\n1040451 20 3 3 23 27.10 10.30\nmp- 629.35 ± 102.84 ±\n2 9 3.59 0.000 Li B S 0.0165\n532413 5 7 13 63.38 22.50\nagm00328 381.37 ± 76.60 ±\n3 105 2.09 0.021 Li Ge(PS ) 0.0156\n2504 10 6 2 51.93 19.55\nmp- 314.62 ± 50.73 ±\n4 2 2.49 0.020 Li P S 0.0178\n641703 7 3 11 18.74 1.43\nmp- 300.41 ± 46.25 ±\n5 105 2.27 0.035 Li Sn(PS ) 0.0156\n696123 10 6 2 17.32 1.64\nmp- 298.29 ± 46.02 ±\n6 62 2.15 0.078 Li NbS 0.0117\n769048 3 4 15.32 6.44\nmp- 273.52 ±\n7 216 2.30 0.083 Li PS Cl 0.0243 0.00 ± 0.00\n985592 6 5 0.00\nmp- 263.98 ±\n8 70 2.30 0.070 Li Cr C SO 0.0141 1.80 ± 0.00\n771912 4 2 4 16 10.42\nmp- 253.08 ±\n9 216 2.14 0.051 Li PS Br 0.0162 0.00 ± 0.00\n985591 6 5 0.00\nmp- 224.56 ± 20.67 ±\n10 33 2.08 0.033 Li PS 0.0119\n1211324 7 6 42.84 5.66\nmp- 189.82 ±\n11 70 2.05 0.076 Li Bi C SO 0.0128 1.51 ± 0.00\n768991 4 2 4 16 25.98\nagm00328 167.51 ± 17.94 ±\n12 105 2.41 -0.008 Li Si(PS ) 0.0186\n2544 10 6 2 16.24 7.20\nmp- 166.70 ±\n13 218 2.42 0.069 Li TiS 0.0196 3.92 ± 2.97\n766600 4 4 18.89\n6\nagm00323 95.80 ±\n14 141 2.02 0.077 Li CuS 0.0165 1.76 ± 1.64\n2549 3 2 12.33\nmp- 65.66 ±\n15 33 2.51 0.026 Li GeS 0.0220 0.70 ± 0.14\n1222582 4 4 0.89\nmp- 50.37 ±\n16 33 2.70 0.063 Li TiS 0.0184 0.93 ± 1.82\n766575 8 6 24.03\nagm00323 22.39 ±\n17 7 4.74 0.041 LiH (SeO ) 0.0251 0.00 ± 0.00\n0254 3 3 2 0.06\nmp- 17.78 ±\n18 1 2.08 0.028 LiFe(SO ) 0.0135 1.41 ± 0.00\n849740 4 2 0.00\nagm00323 16.33 ±\n19 62 4.39 0.064 CsLiSO 0.0122 0.00 ± 0.00\n5290 4 0.00\nagm00221\n20 82 2.01 0.030 LiCdPSe 0.0246 4.14 ± 0.00 0.00 ± 0.00\n9141 4\nmp- LiMgCr (SO )\n21 1 2.43 0.009 3 4 0.0241 0.00 ± 0.00 0.00 ± 0.00\n769554\n6\nmp-\n22 4 4.06 0.068 LiSbCSO 0.0220 0.00 ± 0.00 0.00 ± 0.00\n754902 7\nmp- Li Cr Ni (SO\n23 1 2.25 0.028 3 13 3 0.0202 0.00 ± 0.00 0.00 ± 0.00\n695469 )\n4 24\nmp-\n24 1 2.32 0.011 LiZnCr (SO ) 0.0193 0.00 ± 0.00 0.00 ± 0.00\n769549 3 4 6\nmp-\n25 4 3.75 0.070 LiBiCSO 0.0174 0.00 ± 0.00 0.00 ± 0.00\n770759 7\nmp-\n26 1 2.33 0.009 LiCr Ni(SO ) 0.0179 0.00 ± 0.00 0.00 ± 0.00\n766088 3 4 6\nagm00234\n27 164 2.04 0.072 LiS Sm 0.0124 0.00 ± 0.00 0.00 ± 0.00\n2398 2\nWe thank Reviewer #3 again for these two suggestions, which are important for the\nimprovement of our work.\nList of changes\n1. Lines 283–287 and 312: Results of high-throughput screening have been updated in\nFig. 5 and related analysis; Supplementary Table 9 (line 280) has been revised\naccordingly.\n2. Lines 303–305: Details of MSD calculation methodology have been updated.\n3. Supplementary data (lines 389–394): Descriptions of NNMD simulations and\nassociated MSD analysis have been revised."}, {"round": 3, "reviewer_comments": {"reviewer_3": "The revised methodology and accompanying data are now presented with sufficient rigor to fully substantiate the paper’s\nconclusions. In light of these improvements, I recommend the manuscript for publication."}, "author_response": ""}]

Decision letter