High-throughput screening of sulfur-based silver superionic conductors

Abstract

Superionic conductors (SICs) are attracting significant research attention for their potential applications in all-solid-state batteries. A key challenge in developing such SICs is the rational design of sulfide structural frameworks that enable rapid ion transport. Compared with the conventional trial-and-error approach, theory-driven computational methods offer powerful tools for accelerating materials discovery. In this study, we use the largest cavity diameter (LCD) as a descriptor and conduct high-throughput screening of sulfur-based silver compounds using the Materials Project (MP) database to identify promising superionic-state materials. Machine-learning force-field (MLFF)-accelerated molecular-dynamics (MD) simulations are performed on the selected structures, yielding seven candidate materials, including several previously reported systems. Furthermore, the superionic-state transition temperatures of these compounds are 300 K or higher. Notably, the LCD value is shown to be inversely proportional to the superionic-state transition temperature. This work establishes a quantitative structure–property relationship governing superionic-state transitions and provides a framework for future computational investigations of superionic materials.

Graphical abstract

Keywords

high throughput / largest cavity diameter / Ag superionic conductor / ion diffusion

1 Introduction

Research on superionic conductors (SICs) has been continuously expanding, driven by two core factors. First, there is an urgent demand for the development of next-generation energy storage devices, namely solid-state batteries [1−6]. In the field of solid-state batteries, research on the superionic behavior of lithium-based systems is the most extensive. Certain lithium-containing sulfides, such as Li₁₀GeP₂S₁₂ (LGPS) and LiTi₂(PS₄)₃, exhibit room-temperature ionic conductivity (σ) exceeding 0.01 S/cm [7, 8]. Second, increasing efforts have been devoted to exploring ionic transport phenomena within planetary interiors [9−12]. Studies of superionic states in planetary interiors have enabled the quantification of insulating ice, diverse superionic phases, and liquid water in ice giants [12], and have provided new insights into the simulation of solar system and exoplanetary gas giants [9−11]. Nevertheless, current research primarily focuses on high-efficiency energy storage and utilization. Sulfides deliver ionic conductivity comparable to liquid electrolytes, originating from flexible crystal lattices that facilitate rapid ion transport. Additionally, their excellent ductility enables easier fabrication of sulfides into thin films and intricate microstructures compared with oxide counterparts. Furthermore, sulfide materials generally form low-resistance interfacial contacts with metallic lithium and electrode materials, which is favorable for the development of high-performance all-solid-state batteries. Beyond the aforementioned merits, sulfides feature greater structural design flexibility, lower lattice energy, and strong interfacial interactions, rendering them promising candidates for all-solid-state battery applications [13].

Compared with lithium, silver possesses a highly polarizable electron shell and a suitable ionic radius, making it an excellent medium for fast ion transport [14]. Meanwhile, silver is recyclable, thereby offering superior cost-effectiveness and lower resource consumption than lithium [15]. It also exhibits better chemical stability and temperature tolerance, qualifying it as a reliable candidate for battery systems applied in high-safety scenarios such as military and aerospace fields [16, 17]. The superionic behavior of silver-based materials has been widely investigated, with the earliest discovery dating back to the identification of superionic characteristics in AgI decades ago [18]. Among various silver ionic conductors, α-AgI has attracted substantial research attention. It undergoes a superionic phase transition at T = 420 K under atmospheric pressure, accompanied by a body-centered cubic structural transformation and a three-order-of-magnitude increase in ionic conductivity. Early studies on AgI revealed numerous distinctive physical phenomena. The Parrinello group successfully simulated the reversible α⇄β phase transition of AgI via optimized molecular dynamics methods, elucidating the structural transformation mechanism of superionic conductors [19]. Morgan et al. [20] demonstrated that lattice polarity elevates Frenkel defect concentrations by modulating defect distribution, providing a new strategy for achieving high room-temperature ionic conductivity under non-equilibrium conditions. Recently, Hajibabaei et al. [21] reported that symmetry breaking in the superionic phase of AgI induces competitive phase transitions between tetrahedral and body-centered cubic structures, with an energy difference of only several millielectronvolts. The distribution of silver ions is determined by the polymorph of the parent phase and exhibits nanosecond-scale memory effects. Persistent lattice fluctuations at stable temperatures clarify the molecular origin of such memory behavior and deepen the fundamental understanding of superionic states.

In addition, research on sulfur-based silver superionic materials is ongoing. Previous investigations have covered the rock-salt structured Ag₄Sn₃S₈ [22], superionic behaviors of silver chalcogenides (Ag₂X, X = S, Se, Te) [23], and two-dimensional structured α-KAg₃Se₂ and composite AgI-based materials [24]. These silver-based sulfide and selenide systems exhibit prominent ionic conductivity. For example, Ag₂ZrCl₆ achieves a room-temperature ionic conductivity of approximately 4 × 10⁻³ S/cm [25], while Ag₇P₃Se₁₁ shows a higher value of 13.98 × 10⁻³ S/cm [26]. Fundamental studies on silver ion diffusion, combined with the rapid development of data-driven material design strategies, have significantly advanced superionic material research, particularly for high-throughput material screening via structural descriptors [27−33]. For instance, Muy et al. [34] adopted the phonon band center as a descriptor to screen high-performance lithium-ion conductors. Zhao et al. [35] constructed a hierarchical encoding descriptor based on crystal structural features and established a machine learning (ML) model to predict ion migration activation energy (E) via partial least squares regression. Similarly, Kahle et al. [36] extracted electronic structural features as descriptors and built a predictive model for lithium-ion diffusion coefficients to accelerate the screening of solid-state electrolyte (SSE) materials. The reliability and universality of these structural and electronic descriptors have been well validated, motivating further exploration of the superionic characteristics of silver-based materials.

In this study, a high-throughput screening strategy is employed to explore the correlation between pore structural features and superionic transition temperature. A series of sulfide candidates with low superionic transition temperatures and high ionic conductivity are selected from the Materials Project (MP) database. The present study systematically investigates the inherent correlation between ion diffusion pore dimensions and superionic transition behaviour. Quantitative analysis of such structure−property relationships enables the establishment of an LCD-based characteristic descriptor, which deepens the fundamental understanding of Ag migration mechanisms. This descriptor is further employed for the screening of potential materials exhibiting low superionic transition temperatures. Subsequently, density functional theory (DFT) calculations and machine learning force field (MLFF)-accelerated molecular dynamics (MD) simulations were performed to evaluate the ion diffusion behaviors of screened candidate materials. The integration of the proposed LCD descriptor with high-throughput screening methodologies facilitated the successful identification of superionic materials exhibiting optimal overall performance.

2 Computational methods

All first-principles calculations in this work were performed using the Vienna Ab Initio Simulation Package (VASP), which implements density functional theory (DFT) [37] and is widely used for diverse computational analyses [38−45]. The Perdew, Burke, and Ernzerhof (PBE) generalized gradient approximation (GGA) was used for the electronic exchange-correlation function [46]. The electron-ion interactions were used utilizing the all-electron projector augmented wave (PAW) method [47]. This method has successfully predicted structures of various systems ranging from elements to binary and ternary compounds [48−56]. A plane-wave cutoff energy of 700 eV and a k-point mesh with a minimum spacing of 0.2 Å were set for further geometry optimization and self-consistent calculations. The energy was considered converged when the free energy and band structure energy variations between the two steps were less than 10⁻⁶ eV. Additionally, we considered van der Waals interactions (VDW), choosing the DFT-D3 method with a zero-damping function [57]. To account for strong electronic correlations within 3d orbitals, the Dudarev-type DFT+U formalism (LDAUTYPE = 2) was adopted [58]. An effective Hubbard U value of Fe 4.0 eV (Cr 3.0 eV) was applied exclusively to 3d states, while Ag and S orbitals were not subjected to on-site correlation corrections (LDAUU = 0 eV for Ag/S). The exchange parameter J was set to 0 eV for all atomic species. Spin polarization was enabled (ISPIN = 2), and initial magnetic moments of 12 μB were assigned to each Fe atom, with zero initial magnetic moments placed on Ag and S sites.

A wide range of approaches have been developed for investigating ion transport, with promising results. Monte Carlo simulations serve as an effective modelling tool to probe lithium-ion distribution and hopping, producing physically meaningful ion trajectories and diffusion pathways [59]. The bond-valence site energy (BVSE) method enables rapid calculation of the activation energy (E_a) for the migration of lithium ions in cubic Li-argyrodite. The calculated value serves as training data for the machine learning model [60]. Effective medium theory or the combination of effective medium theory (EMT) and random resistance model (RRM) is used to predict the overall ionic conductivity of composite solid electrolytes [61]. Combining geometrical analysis with the bond-valence method, the initial and final positions of Li₃PS₄ were first identified by crystal-structure analysis via Voronoi decomposition, after which the BVSE approach was employed to compute an approximate minimum-energy pathway (MEP) [62]. Multiple factors, including structure, kinetics, disorder, and interfaces, jointly affect ion transport. Elucidating the mechanisms of electrolyte structure, ionic conductivity and transport dynamics helps develop approaches for tuning ionic conduction, which are rooted in the microscopic physics of ion migration [63].

To identify superionic behaviour and unravel diffusion mechanisms, ab initio molecular dynamics (AIMD) simulations were performed for on-the-fly training over a temperature range of 300–600 K. Machine-learned force fields (MLFF) were then employed to run molecular dynamics simulations on a large supercell of 4800 atoms, further validating superionic behaviour [64−66]. A plane-wave cutoff energy of 520 eV. MLFF has found extensive applications in the investigation of phase transitions, thermal transport, and superionic states, among other research areas [67−76]. On-the-fly training is based on molecular dynamics (MD) simulation samples for training structural models. Where feasible, an automated assembly of a dataset is generated and utilized for MLFF generation. In each time step, the force field predicts energy, forces, and corresponding Bayesian error estimates. Simply put, if the error surpasses a predefined threshold, another ab initio calculation is performed, and the reference energy and forces are added to the training dataset. Conversely, in the absence of such an error, the ab initio steps are omitted, and the system is propagated through MLFF predictions. As the force field along the trajectory improves, many ab initio steps can be circumvented, significantly accelerating the speed of MD simulations. Ultimately, the result of the on-the-fly training is a ready-to-deploy MLFF, allowing for the execution of MD simulations on large crystal cells in prediction mode. This entire scheme has been fully integrated into the VASP code. MSDs can be computed from the following expression, known as the Einstein formula:

(1)\begin{array}{rl}& MSD\left(m\right)\\ & ={\displaystyle \frac{1}{N_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{s}}}}{\displaystyle \sum _{i=1}^{N_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{s}}}\frac{1}{N-m}{\displaystyle \sum _{k=0}^{N-m-1}}[r_i(k+m)-r_i(k){]}^2,}\end{array}

where r_i(t) is the position of atom i after t time of simulation. N_particles and N are the total number of atoms and total frames, respectively [77]. Self-diffusivity is closely related to the MSD, Self-diffusivity can be computed from the following expression,

(2)D={\displaystyle \frac{1}{2d}}\underset{t\to \mathrm{\infty }}{lim}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}MSD\left(m\right).

From the MSD, self-diffusivities D with the desired dimensionality d can be computed by fitting the MSD with respect to the lag-time to a linear model. According to this definition, the MSD is averaged over all windows of length m and over all selected particles. The ionic conductivity was calculated according to the Nernst−Einstein relationship,

(3)\sigma ={\displaystyle \frac{n{q}^2}{K_{\mathrm{B}}T}}D,

where n is the mobile ions volume density, q is the ionic charge, and K_{\mathrm{B}} is the Boltzmann constant.

The migration barrier refers to the energy barrier that an ion must surmount in order to move from one position (usually a vacancy or lattice point) to another. This energy parameter is of pivotal significance in the context of ion migration, exhibiting a close correlation with the conductivity, diffusion coefficient, and overall properties of materials. In the classical diffusion model, the migration of ions follows the Arrhenius relation:

(4)D=D_0\exp \left(-{\displaystyle \frac{E_a}{K_{\mathrm{B}}T}}\right),

where D is the diffusion coefficient, D₀ is the prefactor (related to the microstructure of the material), E_a is the migration barrier (or activation energy), K_B is the Boltzmann constant, and T is the temperature.

3 Results

3.1 The advantages of largest cavity diameter

Figure 1(a) displays the parameters of the largest included sphere and largest free sphere. A larger LCD is found to correlate with wider internal ion-migration channels, which facilitates ion diffusion. We further examined high-performance superionic materials and verified the favourable LCD values reported for these systems in prior work. Given their unique advantages, both Li- and Ag-based superionic conductors have garnered substantial research interest. Typical representatives of Li-based superionic systems, including Li₁₀GeP₂S₁₂ and Li₄GeS₄, deliver a room-temperature ionic conductivity above 10⁻² S/cm. These materials are compatible with high-voltage positive electrodes and serve as core candidate materials for solid-state lithium batteries. Nevertheless, they suffer from high sensitivity to moisture and oxygen, accompanied by abundant interfacial side reactions during service. For the Ag-based system represented by AgI and Ag₄Sn₅S₈, well-established ionic conduction mechanisms can be achieved at ambient temperature, yielding a high ionic conductivity of 10⁻¹ S/cm. Such Ag-based superionic materials are widely applied in sensors and electrochemical switches. In this work, we calculated the LCD values of these extensively investigated superionic structures. The results reveal that all calculated LCD values exceed 2 Å, confirming the large structural channel sizes of these systems (See the supplementary material for the section “Supplementary Note 2”). Sodium superionic conductors have achieved rapid progress in recent years, yielding many high-performance materials such as novel sulfide and halide systems. Incorporation of highly polarizable anions or defect-engineering strategies enables ultra-low activation energies (< 0.25 eV) and wide electrochemical windows (> 5 V) under mild operating conditions. Furthermore, we performed calculations on these novel Na-based superionic conductors. All compounds exhibit LCD values above 2 Å, validating LCD as a viable screening descriptor for Ag-based superionic conductors. Favorable LCD values facilitate ion migration through wider channels. Increasing the cavity diameter can substantially mitigate the steric blocking from the lattice skeleton toward ion transport. Proper tuning of LCD dimensions helps lower the ion-migration energy barrier. From an energetic viewpoint, ion diffusion inherently requires overcoming characteristic energy barriers. With an adequately enlarged cavity diameter, both the deformation of ions moving in and out of cavities and their interactions with surrounding atoms become less pronounced. It is evident that LCD exerts a notable influence on the distribution of ion diffusion pathways. The large interconnected cavity structure enables multiple potential migration routes for ions. Consequently, ion diffusion becomes more flexible, and the overall transport continuity can be preserved despite local structural defects or slight structural perturbations.

3.2 The relationship between LCD and transition temperature

As shown in Fig. 1(b), we retrieved 130 silver sulfide compounds from the Materials Project database via the pymatgen package. After sequential screening and computational simulations, seven structures were confirmed to possess superionic states [see red solid pentagrams in Fig. 2(a)].

In the course of the present study, a number of crystal structures were identified that were found to be identical to those previously reported compounds (e.g., Ag₇NbS₆, Ag₇TaS₆ and AgCrS₂) [78, 79]. Furthermore, our simulation results show good agreement with previously reported data for these well-known compounds. After screening, the LCD values of all structures were sorted in descending order, as shown in Fig. 2(b). Of all investigated materials, AgCrS₂ presents the maximum LCD value of 3.55 Å. Furthermore, the superionic state transition temperature of AgCrS₂ is approximately 300 K, a finding that aligns with the conclusions drawn from prior research [79]. At LCD values of 2.2–2.3 Å, the number of candidate structures reaches its maximum, with only minor variations in LCD values. Despite negligible changes in LCD, the superionic transition temperatures of these structures show regular trends, as seen from the brown lines in Fig. 2(b). A progressive decrease in LCD corresponds to an increase in the superionic transition temperature, indicating an inverse relationship between these two parameters. Subsequent MLFF-accelerated molecular dynamics simulations were used to validate the hypothesis that the superionic transition temperature rises gradually with decreasing LCD. Although several structures show sharp temperature fluctuations, the overall trend remains evident. These results underscore the importance of well-tuned ion-diffusion channels for developing superionic materials operating at ambient temperature.

3.3 Structure characteristics

Crystal structures identified via high-throughput screening fall into two main categories: layered frameworks and complex three-dimensional structures (Supplementary Fig. S2). Layered structures generally possess large LCD values originating from interlayer van der Waals forces or ionic bonding, favoring Ag-ion diffusion. Weak interlayer interactions reduce the diffusion barrier, which in turn enhances ion migration [80]. Nonetheless, this structural feature constrains the three-dimensional diffusion capability of silver ions. When ample channels exist between sublattices, three-dimensional diffusion can be realized with improved efficiency. In contrast, Ag ions in complex three-dimensional frameworks adopt high-coordination environments enforced by ionic bonding. These effects reduce the LCD and elevate the migration barrier, significantly suppressing ion diffusion. Accordingly, it remains challenging to discover materials with favourable room-temperature Ag-ion transport in such three-dimensional structures.

The largest cavity diameter (LCD) is a structural parameter characterizing the bottleneck aperture for Ag⁺ migration. It arises from the hierarchical coupling of multiple structural parameters, instead of the Ag coordination number in isolation. Framework topology and channel connectivity are the primary factors governing the feasibility of continuous long-range ion transport. Three-dimensionally interconnected tetrahedral skeletons form complete percolative void networks. By contrast, layered stacking isolates intralayer cavities through van der Waals gaps, breaking through-going diffusion pathways even when local LCD values are large. Ag coordination number and polyhedral packing constitute key local geometric regulators, whose effects hold only within a given class of topological frameworks. Low-coordinated Ag occupies a smaller lattice volume, expanding interstitial voids and increasing LCD. By contrast, dense polyhedral stacking suppresses cavity expansion regardless of Ag coordination. Consequently, coordination number and LCD exhibit an indirect correlation instead of a direct causal link. Metal-sulfur bond length acts as a secondary modulator that adjusts cavity size via changes in polyhedral volume. Small static sublattice distortions induce only modest local fluctuations in LCD and do not alter its overall magnitude. Layered AgCrS₂ exemplifies this behaviour: it has low-coordinated Ag and the largest LCD of all investigated samples, yet suffers a very high migration barrier originating from two-dimensional topological confinement. For three-dimensional sulfides with unobstructed diffusion networks, the synergistic effects of coordination environments, polyhedral arrangements and bond lengths maintain a robust inverse correlation between LCD and superionic transition temperature across all screened candidates.

ELF (Supplementary Fig. S3) calculations show no obvious localized electron maxima between Ag and S ions within all screened structures, which eliminates the existence of strong localized covalent bonds featuring concentrated electron accumulation at Ag–S bonding sites. Nevertheless, the lack of well-defined ELF peaks does not constitute sufficient evidence to dismiss weak delocalised polar-covalent mixing superimposed onto the prevailing ionic interaction. Because ELF only resolves discrete localised electron basins, it cannot quantitatively separate purely ionic bonding from subtle delocalised covalent contributions. Given the high polarizability of Ag⁺ cations and considerable electronegativity of sulfide anions, Ag–S bonding is dominated by ionic character, accompanied by small diffuse polar-covalent components; these bonds are thus neither fully ionic nor purely covalent.

We selected one material, the complex three-dimensional structure, for analysis, namely Ag₉AlS₆. As shown in Fig. 3(a), Ag₉AlS₆ is a complex three-dimensional structure with space group P1. In this structure, the Al−S atoms are mutually bonded to form the sublattices of the crystal.

3.4 MLFF simulation results

We conducted MLFF-accelerated molecular dynamics (MD) simulations on the selected structures to obtain their superionic transition temperatures and ionic diffusion data. The Ag superionic state was described in detail through representative structures (Ag₉AlS₆). As demonstrated in Fig. 3(b), the mean square displacement (MSD) of the structures unequivocally demonstrates that Ag₉AlS₆ exhibit superionic state under the temperature driven conditions. We further analyzed the MSD results for other structures, as presented in Supplementary Fig. S4. The sublattices of both structures remain dynamically balanced upon temperature changes, and no obvious ion diffusion tendency is observed. Meanwhile, the MSD curves confirm that the superionic transition temperature of silver ions in these structures is 400 K. As shown in Fig. 3(b), the displacement of silver ions from their initial positions increases continuously over the simulation. Concurrently, elevated temperatures reduce the time required for Ag ions to traverse a given distance, thereby enhancing the diffusion rate. As demonstrated in Figs. 3(c) and (d), the trajectories of Ag₉AlS₆ at 600 K for the initial position of Ag ions and the complete diffusion are presented, respectively. The diffusion of this structure is three-dimensional, and the diffusion trajectory of Ag ions only extends within the S ion gaps, with no diffusion occurring in the sublattices.

In order to achieve a more profound comprehension of the local characteristics exhibited by structures during the diffusion process, the calculation of the radial distribution function (RDF) was calculated. As illustrated in Fig. 4, the radial distribution function g(r) of Ag₉AlS₆ is presented. The simulation results of g(r)_Total for Ag₉AlS₆ demonstrate that at differing temperatures, the position of the primary peak is approximately 2.5 Å [Fig. 4(a)], corresponding to the mean nearest distance between Ag and S. As demonstrated in Supplementary Fig. S5(u), at a temperature of 300 K, the local environment between Ag and S exhibits a degree of order, with a peak height of approximately 4.1. As the temperature rises, the peak undergoes a gradual decrease, indicative of a concomitant weakening of local order. It is evident that the second maximum of g(r)_Total is positioned at approximately 3.8 Å, corresponding to the mean nearest distance between Ag and Al. As demonstrated in Fig. 4(c), the initial peak of Ag−Al at 300 K is estimated to be around 3.8 Å. However, in the temperature range of 400–600 K, the peak intensity decreases, and the curve broadens, indicating increased structural disorder. This phenomenon marks the transition of silver ions into the superionic state, accompanied by active ion diffusion. As shown in Fig. 4(b), Ag ions enter the superionic state at elevated temperatures. However, the peak of g(r)_Ag−Ag remains largely unaltered with respect to temperature, and the broadening exhibits minimal variation. This result indicates that Ag ion diffusion retains relatively local structural order. This supports the hypothesis that high ion content sustains such local ordering. For the Al–S sublattice [Fig. 4(d)], both peak position and peak width exhibit negligible temperature-dependent changes. Accordingly, this sublattice retains high structural order across the investigated temperature range. This observation aligns with its structural stability during Ag-ion diffusion.

LCD provides ample lattice voids, which widen Ag⁺ migration pathways, mitigate spatial repulsion between ions and neighbouring atoms, and reduce lattice distortion during diffusion. These characteristics enable rapid Ag⁺ transport through sulfur-coordinated interstitial sites. At the same time, heating triggers the superionic transition of Ag⁺, yet diffusion does not evolve into complete structural disorder. Ion transport proceeds within the locally ordered void network linked to large LCD values. The nearly unchanged peak position in the Ag–Ag radial distribution function confirms that Ag ions preserve well-defined local structural order during migration. This cooperative wide-channel plus local-order mechanism effectively reduces the diffusion barrier, reconciling high ionic conductivity with structural stability.

3.5 Diffusion path and energy barrier

Here we studied the diffusion paths of two-dimensional and three-dimensional structures. Molecular dynamics simulation was carried out on the material, and its motion path was shown in Fig. 5. The detailed Ag ion migration pathways of Ag₉AlS₆ are illustrated in Supplementary Figs. S6(a)−(c). The migration pathways of the Ag ions tend to diffuse along the interstitials surrounding the S ions [As shown in Figs. 5(a) and (b) and Supplementary Figs. S6(a)−(c)]. In Ag₉AlS₆, the distance between the sulfide ion and the sublattice exceeds 4 Å, thus ensuring the presence of sufficient vacancies. The ionic probability density (Fig. 6) was constructed through data analysis, which revealed that Ag ions manifest three-dimensional diffusion in Ag₉AlS₆ [Fig. 6(a)]. In the layered material AgFeS₂, Ag ions exhibit two-dimensional diffusion behavior within the Ag layers [Fig. 5(c) and Fig. 6(b)]. The migration pathways of Ag ions are thus classified into three distinct categories, as illustrated in Supplementary Figs. S6(d)−(f). The Ag⁺ migration pathways examined in this work comprise hopping between sulfur-coordinated sites via second-neighbour S atoms, together with trajectories passing near adjacent Fe centres. The Fe–S sublattice offers only limited interlayer interstitial volume, hindering Ag⁺ transport and in turn suppressing cross-layer diffusion as well as the overall ionic conductivity.

The Arrhenius equation was applied to determine the energy barriers of the investigated structures. As presented in Table 1, the majority of energy barriers are on the order of 0.1 eV, well explaining the low superionic transition temperatures of these materials. The findings further demonstrate the practicability of adopting LCD as a descriptor to screen superionic conductors. This strategy both validates previously reported structures and identifies new promising candidates. The approach is valuable for investigating superionic solid-state batteries as well as ion-transport behaviour within planetary interiors. MSD-derived data were processed using formula (2) to compute the ionic conductivity at each corresponding temperature (Table 2). It was established that in structures exhibiting a superionic state at room temperature, the ionic conductivity is observed to exceed 10⁻³ S/cm. For instance, the ionic conductivity of Ag₆NbS₆ reaches 0.335 S/cm at room temperature, which is related to its low energy barrier and large LCD. Ag₉AlS₆ also achieves 0.167 S/cm at 300 K. As the temperature rises, the ionic kinetic energy increases and the migration ability enhances, so the ionic conductivity usually increases with the rise of temperature. The ionic conductivities of Ag₉AlS₆ from 400 to 600 K are 0.312, 0.699, and 1.112 S/cm, respectively, while those of AgFeS₂ are 0.088, 0.227, and 0.374 S/cm, respectively. Improving the ionic conductivity of materials, especially at ambient conditions, represents a key research priority. Prospective research avenues involve discovering new materials with high ionic conductivity, designing novel architectures to optimise ion-migration pathways, and combining computational simulations with experimental characterisation to lay theoretical foundations for conductivity enhancement.

A larger LCD corresponds to an expansive, interconnected void network in the crystal lattice. These structures furnish plentiful three-dimensional migration pathways for Ag⁺, allowing ions to hop readily through sulfur-coordinated interstices and bypass high-energy sublattice domains. Ag₉AlS₆ serves as a representative example. The large LCD of this compound results in a three-dimensionally interconnected diffusion path, with a wide and continuous probability density distribution of Ag ions. This significantly reduces local hopping resistance. Consequently, the diffusion barrier of Ag₉AlS₆ is only 0.134 eV, which is much lower than that of complex three-dimensional structures with smaller LCD. Conversely, materials with smaller LCD feature narrow cavities and disconnected channels, forcing Ag⁺ to traverse highly distorted regions. This yields restricted diffusion paths and higher migration barriers. It can thus be concluded that LCD exerts two-fold effects on Ag⁺ transport. Firstly, it governs the diffusion dimension and path diversity. Secondly, it directly lowers the migration barrier by mitigating local geometric constraints, which in turn improves the overall ionic conductivity.

To contextualize the structural and transport characteristics of our screened silver sulfide candidates, we perform systematic benchmarking against five representative well-studied silver-based superionic systems. First-principles calculations by Sun et al. show that classic BCC-type binary silver chalcogenides including α-AgI, β-Ag₂S and α-Ag₂Se adopt 4–6-fold high-coordinated interstitial Ag sites, yielding limited maximum cavity diameters of 2.1–2.2 Å with superionic phases only stable above 406–450 K [81]; further neutron diffraction studies by Hull et al. and molecular dynamics simulations by Rino et al. confirm that these BCC-framework materials only support one-dimensional Ag⁺ migration along crystallographic directions without fully interconnected 3D transport pathways, and Ag₂S/α-Ag₂Se are narrow/zero-bandgap mixed conductors that suffer from intrinsic electronic leakage defects [82−84]. Makiura et al. stabilized the superionic α-phase of AgI to room temperature via polymer coating on 10–11 nm nanoparticles, but the room-temperature conductivity remains only 1.5 × 10⁻² S/cm, and the performance relies on organic surface modification rather than intrinsic bulk stability [85]. For the ternary rock-salt Ag₄Sn₃S₈, its dense SnS₆ octahedral framework restricts cavity expansion and transport efficiency, with a superionic transition temperature around 390 K and room-temperature conductivity in the order of 10⁻³ to 10⁻² S/cm. In comparison, our Al/Nb/Ta-based 3D tetrahedral framework sulfides feature 2–3-fold low-coordinated Ag sites that release sufficient lattice space to form interconnected cavities with LCD values of 2.2–3.0 Å, constructing fully percolative 3D transport channels that enable intrinsic superionic behavior below 300 K without any extrinsic modification; four candidates deliver room-temperature ionic conductivity exceeding 0.1 S/cm as wide-bandgap pure ionic conductors, eliminating the mixed-conduction drawback of conventional binary systems. For layered AgCrS₂, its record-high intralayer LCD of 3.55 Å cannot be converted into efficient three-dimensional ion transport due to van der Waals interlayer blocking, which restricts diffusion to in-plane two-dimensional paths and results in a high migration barrier of 0.509 eV; this low-dimensional confinement effect aligns well with the transport behavior of reported 1D-confined superionic systems, confirming the physical validity of our simulation results. This systematic comparison demonstrates that the 3D rigid tetrahedral framework silver sulfides identified in this work are a distinct new class of superionic materials separate from existing BCC-type binary and close-packed rock-salt ternary silver conductors, with prominent structural and performance advantages.

4 Discussion

The candidates selected for detailed analysis consist mainly of layered and intricate three-dimensional frameworks. This work confirms that LCD constitutes an effective descriptor for screening superionic materials. Our results reveal an inverse correlation between LCD and the superionic transition temperature. Notably, several of these structures deliver high ionic conductivity (> 10⁻³ S/cm); further work is therefore needed to assess their practical application potential. The migration-energy barrier directly reflects the resistance opposing Ag-ion hopping, with its magnitude governed by a confluence of several physical factors. Dense atomic packing, low defect concentrations and strong interionic interactions all act to increase this barrier. Accordingly, reducing the migration barrier represents a core strategy for promoting ionic transport and optimising the bulk material performance. Reducing energy barriers is essential for enhancing ionic conductivity, catalytic efficiency and material reactivity across solid-state-electrolyte research, catalyst development and broader materials-science fields. Although high temperatures can overcome these constraints, they impose substantial limitations on the practical applicability of candidate materials. Introducing defects represents a well-established strategy for barrier lowering, which proves especially effective for systems intrinsically lacking vacancy sites. Nevertheless, defects may compromise structural integrity, particularly under high-temperature conditions, and can ultimately trigger material degradation. As demonstrated in our previous work, Ag-related defects strongly affect the superionic transition temperature of AgAlO₂ and AgFeO₂. Nevertheless, defect incorporation yields heterogeneous material properties, which can degrade performance, especially for high-performance applications. It is therefore essential to define a reasonable window for material modification. Beyond defect chemistry, sublattice architecture and its associated parameters also govern the superionic transition temperature. Sublattice-distortion magnitude, varied structural motifs and other physical factors collectively tune superionic-transition behaviour. Our MLFF simulations capture prominent structural fluctuations in layered phases. We infer that such structural perturbations reshape internal channel geometry, homogenise electron distribution and reduce migration barriers, thereby facilitating Ag-ion diffusion. In other structural configurations, sublattices vibrate within a stable amplitude window, which also promotes homogeneous electron distribution. Furthermore, the resulting varied sublattice distortions can yield widened Ag-ion migration pathways. In the present work, we have investigated these potential contributing factors. Although we used MLFF simulations and first-principles calculations to characterize ion-migration behaviour, such theoretical models cannot capture the full complexity of real materials, including surface effects and macroscopic structural features. These factors may exert a substantial influence on ionic conductivity in practical applications.

We conducted spin-polarized DFT+U single-point energy barrier calculations for the layered magnetic sulfides AgFeS₂ and AgCrS₂. The spin-resolved electronic states revealed a significant spin splitting in the Fe 3d orbitals, confirming that this material has a stable intrinsic magnetic ordered ground state. However, the spin splitting in the Cr 3d orbitals was not as strong (Supplementary Fig. S8). Additionally, for these two structures, the influence of magnetism is different. For AgFeS₂, the energy barrier increased by 0.15 eV, while for AgCrS₂, there was no significant change. Combining the MSD obtained from the MLFF dynamics, it can be seen that in a weak magnetic field environment, spin-phonon coupling can increase the ionic transition resistance, and magnetism only acts as a secondary regulating factor for transport properties. The two-dimensional structure directly cuts off the three-dimensional ion diffusion channels, which is the core reason for the insufficient transport performance of layered materials. Combined with the relevant literature reports of the MCrX₂ system [86], the strong magnetic order at the Cr site will significantly enhance the spin-phonon coupling effect, further raising the ion migration barrier. In the organic ionic plastic crystal reported by Qian et al. [DEIm][Ni(mnt)₂], the magnetic anion [Ni(mnt)₂]⁻ forms a one-dimensional spin chain [87]. Its magnetic bistability is coupled with the negative thermal expansion of the lattice, which in turn affects the migration path of cations and the ionic conductivity. Overall, the comparison of these results with the literature indicates that the magnetic effect only has a secondary regulatory effect on the ion migration barrier, while the crystal structure of the material itself is the most significant factor.

The LCD descriptor represents a purely geometric-topological parameter computed from the static anionic framework upon removal of all Ag atoms. It depends exclusively on the stacking arrangement of the crystal framework, interlayer spacing and Ag-coordination environments, and is entirely independent of electronic-spin states. For nonmagnetic three-dimensional-framework candidates, which constitute the majority of our screening library, magnetic effects do not come into play. Accordingly, they exert no impact on individual LCD values nor on the observed inverse correlation between LCD and the superionic transition temperature (T_c). In the case of layered magnetic AgFeS₂ and AgCrS₂, their intrinsic cavity sizes remain geometrically invariant regardless of spin polarization. Nevertheless, spin-order-triggered spin–phonon coupling acts as a secondary factor that modulates the migration energy barrier of Ag⁺ within the fixed LCD transport channels.

While LCD constitutes a core parameter for characterising ionic-transport behaviour in three-dimensional silver-sulfide superionic conductors, its valid scope and intrinsic limitations as a standalone descriptor remain poorly elucidated. Accordingly, we decoupled and analysed four key physical factors — coordination number, structural dimensionality, magnetism, and sublattice distortion — to identify the specific physical regimes where LCD dominates ionic transport. Structural dimensionality is the decisive constraint for LCD validity. For 3D frameworks (Ag₉AlS₆, Ag₇NbS₆, Ag₆NbS₆), rigid Al/Ta/Nb-S sublattices form fully connected diffusion networks, and LCD exhibits a stable inverse correlation with superionic transition temperature and migration barrier. In layered materials like AgCrS₂ and AgFeS₂, however, weak van der Waals gaps block cross-layer Ag⁺ transport. Despite AgCrS₂ having the largest LCD (3.55 Å), 2D confinement restricts only in-plane ion migration, yielding an unusually high barrier of 0.509 eV. This anomaly arises from dimensional limitation rather than failure of the LCD rule. The Ag-coordination number is strongly coupled to LCD: low-coordinated Ag centres expand lattice voids and increase LCD, while high-coordination environments compress the available free volume. Nevertheless, coordination acts only as an auxiliary factor and cannot counteract the dominant effect of structural dimensionality. Magnetism and sublattice distortion act as secondary modulating factors without reversing the LCD core trend. In magnetic AgCrS₂, Cr³⁺ spin-phonon coupling further elevates migration barriers, whereas non-magnetic 3D sulfides maintain consistent LCD-temperature correlations. Rigid sublattices in our 3D samples experience minimal thermal distortion and preserve stable large cavities; severe lattice distortion conversely shrinks effective LCD and hinders diffusion.

We summarise the applicable conditions under which LCD acts as the dominant descriptor. First, the material should possess continuous three-dimensional percolative diffusion channels free of two-dimensional interlayer barriers. Second, it should feature rigid sublattices with low distortion. Third, transition-metal species should be present within the structure. Fourth, the intrinsic vacancy concentration should be moderate, such that vacancies do not fill the lattice cavities.

The standalone LCD descriptor exhibits notable limitations. It disregards interlayer connectivity, magnetic-spin coupling, as well as dynamic cavity evolution driven by high-temperature sublattice behaviour and defect rearrangements. AgCrS₂ constitutes merely an extreme limiting case originating from the superposition of two-dimensional confinement and magnetic effects. This outlier does not invalidate LCD for mainstream non-magnetic three-dimensional silver-sulfide superionic conductors. Follow-up efforts will combine LCD with coordination number, sublattice distortion, and magnetic parameters to establish a multi-descriptor model aimed at broad-scope high-throughput screening for sulfur-based silver superionic conductors.

Thanks to their abundant internal cavities and two-dimensional migration pathways, the LCD descriptor shows high applicability for layered structures. By contrast, complex three-dimensional frameworks yield substantially reduced LCD values, which imposes constraints on Ag-ion diffusion. Moreover, the present analysis does not account for real-time effects originating from defects including vacancies and lattice distortion, alongside temperature-driven dynamic fluctuations of diffusion pathways. Future work aims to build a “dynamic-multiscale” LCD framework consisting of three components: real-time coupling between MD-derived cavity dimensions and local parameters such as coordination number and electron density; synergistic tuning of effective apertures through defect-engineering strategies; and extension of the model to iteratively refine theoretical predictions for grain boundaries and electrode interfaces. This strategy will facilitate targeted material design and improvements in device-level performance.

5 Conclusion

Herein, the largest cavity diameter (LCD) was utilized as a screening indicator to perform high-throughput screening on sulfur-based silver compounds, with the objective of exploring superionic solid-state electrolytes. By coupling first-principles calculations and machine learning force field (MLFF) simulations, we conducted an in-depth analysis of the ion diffusion mechanisms and transport characteristics of the selected materials. The experimental and computational results confirm a strong inverse relationship between LCD values and the superionic transition temperature. According to MLFF simulation results, all selected materials deliver a room-temperature ionic conductivity higher than 10⁻³ S/cm. Such characteristics demonstrate their capability to act as high-performance solid electrolytes and confirm the practicability of LCD. The rationality of the proposed descriptor is further verified via RDF analysis, atomic trajectory tracking, and energy barrier calculations. While this study has made considerable theoretical progress, follow-up research is necessary to explore their structural stability and practical application performance. Subsequent research will be dedicated to experimentally verifying the computational results, optimizing structural stability, and elucidating the detailed mechanism underlying the relationship between LCD and the superionic transition temperature. Additionally, efforts will be devoted to developing high-performance ionic conductors and designing new structural configurations to refine ion migration channels. The synergy between computational simulation and experimental testing will lay a theoretical foundation for the improvement of ionic conductivity. Progress in high-throughput screening and material synthesis technologies will enable the fabrication of more creative and applicable ionic conductive materials, which will greatly advance the field of energy storage and conversion.

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