Introduction

Solid-state electrolyte (SSE) materials replacing the traditional flammable organic solvent electrolytes, can effectively combine with lithium metal anode and increase the safety and energy density of all-solid-state lithium ion batteries (ASSLIBs)1,2. For the practical application, lithium ionic conductivity at room temperature of an SSE material should be comparable to liquid electrolytes, more than 1mS cm−1, and SSE materials are superionic conductors. Thus, deeply understanding the physical picture of ion transport in the current ionic conductors is the key to further optimizing and developing lithium superionic conductors as SSE materials. The current understandings of fast lithium diffusion in ionic conductors are mainly from two aspects: crystal structure3,4 (static mechanics) and ion-lattice interaction dynamics5,6. Recently, some computational researches by the ab-initio molecular dynamics (AIMD) simulations show the coupled Li-ployanion dynamics between polyanion rotational motions and cation translational motions would enhance cation diffusion, called the paddle-wheel effect7,8, including 0.75Li2S-0.25P2S59 glass, Li3.25P0.75Si0.25S410, LiBF411 and Na11Sn2PX12 (X = S and Se)12 crystals. These AIMD simulations for lithium or sodium ionic conductors were mainly performed at elevated temperatures (>600 K) to enhance the sampling of ion diffusion events and obtain sufficient diffusion statistics at the time level of ~100 picoseconds (ps)10,12, because the rare ion diffusion events in the low-temperature AIMD simulation would result in high statistical error. On the other hand, extrapolated room-temperature ionic conductivity and diffusivity by the Arrhenius assumption usually have large errors13. As these polyanion rotations and paddle-wheel effect are typically observed in the high-temperature phases with large free volumes and kinetic energies of atom, exploiting polyanion rotation in lithium ionic conductors at low temperatures is still a challenge. A question then naturally arises: whether polyanion rotation and even the paddle-wheel effect can persist down to the common ionic conductors at room temperature?

Today, the ab-initio calculation, molecular dynamics (MD) simulation and machine learning (ML) method are widely utilized for studying and developing advanced battery materials14,15,16. Especially for ML combine the classical computational methods, it is a powerful tool to efficiently predict properties and analyze the complex structure-function relationships of battery materials17,18. For example, Zhao et al. developed a ML prediction model enabling rational design and optimization of Li-argyrodites solid-state electrolytes based on the hierarchically encoding crystal structure-based descriptors19,20, and Wang et al. combined the ab initio calculation and ML method to identify chemical factors affecting reaction kinetics in Li-O2 battery, demonstrating the critical role of disorder degree of LiOH and solvent effect21. MD simulation based on the machine learning interatomic potential, so-called machine-learning molecular dynamics (MLMD) simulation, where energies and forces are directly obtained by solving the Schrödinger equation via the density functional theory (DFT) calculation, is a very powerful tool to extend the simulation time scale from ps level to microsecond (μs) level and simultaneously keeps accuracy at the DFT level22,23. In this work, we first developed the machine-learning interatomic potentials for five lithium ionic conductors, including Li7P3S11, Li10GeP2S12, β-Li3PS4, Li3ErCl6 and Li3YBr6 (Fig. S1), and performed MLMD simulations at the time level of μs to capture polyanion rotation event and explore the paddle-wheel effect in above five lithium ionic conductors at 300 K, providing us a direct physical picture of the relationship between polyanion rotation and cation diffusion in ionic conductors at room temperature. Moreover, by MLMD simulations at low temperatures, we explored the non-Arrhenius behavior in these five lithium ionic conductors, which bridges the gap between experimental and simulated ionic conductivities.

Results and discussion

Performance of MTP potential

There are three major contradictions for reasonably and efficiently applying ML in material community, including contradictions between learning results and domain knowledge, between model complexity and ease of use, between high dimension and small sample data, which were firstly identified by Shi et al.17,18,24. Embedding domain knowledge of material science inside ML method to would effectively reconcile the above three contradictions. In this work, the domain knowledge of crystal structure, bond length and coordination environment of local atom is chosen as the structure descriptors to establish the mapping relationships between structure and energy (force). Fitting MTPs with respect to different levmax values were considered, and the performance of MTPs predicted lattice constant, volume, and mean absolute errors (MAEs) in energies and forces are shown in Figs. S2–S11 in Supporting Information. Comprehensively considering the accuracy of predicted lattice constant and volume, MAEs in energies and forces are regarded to converge at levmax of 10, 12, 10, 18, and 18, respectively for five lithium ionic conductors, as listed in Table 1, and MTPs for the following MLMD were based on these five levmax values. In all cases, MAEs in energy are between 1.02 meV atom−1 and 3.86 meV atom−1, while MAEs in force are less than 0.12 eV Å−1. These MAEs are similar to or less than those of other MTPs fitted in the previous work25,26, showing a remarkable improvement over the traditional interatomic potentials. Our MAEs in energy of the training and test data of Li7P3S11 are larger than the previous work by S.P. Ong27, but our MAEs in force are much close to them, 0.11 vs. 0.09. Additionally, viewed from Table 1, MAEs of the training and test data are generally very similar, demonstrating a little possibility of overfitting potentials. Our fitted MTPs would consistently exhibit high accuracy of reproducing DFT energies and forces at different temperatures. Table 2 compares lattice parameters and volumes for five lithium ionic conductors between MTP relaxations and experimental values. It can be seen that MTPs are generally able to reproduce lattice parameters and make absolute error less than 2.19% compared to experimental values, expect for Li7P3S11. On the other hand, making comparison with the previous work of Li7P3S11 by Ong27, our MTP predicted lattice parameters and volume are much close to them. In all, our MTPs are reliable and well reproduce DFT energies and forces, lattice parameters, and local structures of these five lithium ionic conductors.

Table 1 Mean absolute errors (MAEs) of energy and force predictions for fitted MTPs of five lithium ionic conductors.
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Table 2 Lattice parameters and unit cell volumes of five lithium ionic conductors predicted by the trained MTPs at 0 K, in comparison with the experimental data at room temperature for Li7P3S1145, Li10GeP2S1264, β-Li3PS465, Li3ErCl628, and Li3YBr666.
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Non-linear Arrhenius behavior

Based on the fitted MTPs, we performed low temperature MLMD simulations for five lithium ionic conductors. Figure 1 shows the Arrhenius plots and a summary of the derived activation energies (Ea) for Li ion diffusion. Viewed from the Arrhenius plots (filled markers and solid lines), four lithium ionic conductors, including Li7P3S11, Li10GeP2S12, β-Li3PS4 and Li3YBr6, exhibit the single linear Arrhenius behaviors from 300 K to 500 K, which is the traditional assumption made when extrapolating diffusivity from the high-temperature (>500 K) AIMD simulations to room temperature. While for Li3ErCl6, the non-linear Arrhenius behavior was observed, and the transition between the high-temperature (HT) quasi-linear regime with lower Ea (0.165 eV) and the low-temperature (LT) quasi-linear regime with higher Ea (0.425 eV) occurs at ∼460 K. Therefore, following the traditional assumption of linear Arrhenius regime, directly extrapolating ionic conductivity at room temperature (σ300K) from high-temperature (460–520 K) would cause a significant overestimation from 1.37 to 48.48 mS cm−1. Our MLMD predicted Ea and σ300K of 0.425 eV and 1.37 mS cm−1 for Li3ErCl6 is much close to the experiment determined results of 0.416 eV and 0.31 mS cm−128, demonstrating that the near room temperature MLMD simulations well capture the transitions between quasi-Arrhenius regimes, and increase the accuracy of calculated σ300K. To dig the reason behind the observed transitions between quasi-linear Arrhenius regimes of Li3ErCl6, Li trajectory lines from MLMD simulations near the transition temperature were extracted, as shown in Fig. S12. We noticed the order-disorder transition of Li ion sublattice. Below the transition temperature, Li ions primarily diffuse along c-axis direction, while additional Li ion diffusion paths along a-b planes are enabled above the transition temperature (500 K, Fig. S12b). These activated additional diffusion paths above 500 K increase the dimensionality for Li ion diffusion and decrease Ea from 0.425 eV to 0.165 eV.

Fig. 1: Temperature-dependent Li ion diffusion coefficients by MLMD simulations of five lithium ionic conductors.
figure 1

a Li7P3S11, Li10GeP2S12, Li3YBr6 and β-Li3PS4 show the Arrhenius behaviors at low-temperature ranges of [300 K, 500 K], while b Li3ErCl6 exhibits the non-linear Arrhenius behavior at low-temperatures, and the transition occurs at 460 K. 1μs MLMD simulations for Li7P3S11, c direction projected MSD curves of Li ion at 300 K, and d isosurfaces of the Li+ probability density distribution P (light blue) at 300 K with P = 0.0005 a0−3 (a0 is the Bohr radius).

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Polyanion rotation

To capture sufficient Li diffusion and polyanion rotation events at room temperature, MLMD simulations for five lithium ionic conductors were performed at 300 K as long as 1μs. In Li7P3S11, the maximum amplitude for Li vibration at room temperature is ~0.05 Å, and the mean amplitude is ~0.02 Å, as shown in Fig. S13. Figure 1c shows the total MSD and x-, y-, and z-direction projected MSD of Li ion diffusion in Li7P3S11 as a function of simulation at 300 K. Total MSD of Li is about 10000 Å2 after 1μs MLMD simulation, indicating considerable Li diffusion events in Li7P3S11 at 300 K. The corresponding Li ion diffusion coefficient and ionic conductivity at room temperature are calculated to be 1.40 × 10−7 cm2 s−1 and 14.96 mS cm−1, respectively, much close to the experiment determined results of 1.65 × 10−7 cm2 s−1 and 11.6–17 mS cm−1 29,30 at 300 K. Due to the possible non-Arrhenius behavior, extrapolating room-temperature ion diffusivity and ionic conductivity by the Arrhenius assumption usually may cause large error. However, directly calculating Li ion diffusion coefficient and ionic conductivity from room temperature MLMD simulation would effectively reduce errors. Therefore, the calculated diffusion properties from room temperature MLMD simulation are more accurate and meaningful. We also note that the deviation between experiment value and our calculated Li ion diffusion coefficient of 2.3 × 10−7 cm2 s−1 from 1 ns MLMD simulation is higher than that from 1μs MLMD simulation, demonstrating that extending simulation time to 1μs would significantly reduce the statistical errors of Li ion diffusion. Viewed from MSD data, Li ion diffusion along x-direction is more preferred than y- and z-directions, but it cannot illustrate Li7P3S11 is a 1D ionic conductor at room temperature. In addition, isosurfaces of Li probability density distribution (Fig. 1d) clearly show 3D connected channels for Li ion diffusion. The good Li diffusivity provides sufficient diffusion events for us to explore the paddle-wheel effect in Li7P3S11 at 300 K.

Compared to initial structure (Fig. 2a), significant polyanion rotations were noticed from last structure (Fig. 2b) of Li7P3S11 after 1μs MLMD simulations at 300 K, including P5, P6, P7 and P8. For example, for central P8 atom, S4, S5 and S44 rotate about 120 or 240 degrees along P8-S28 bond axis (C3 local symmetry axis), as depicted in Fig. 2c. During the whole MLMD simulation times, isolated [PS4]3− group does not rotate, and all rotated [PS4]3− tetrahedra belong to a part of the longer-chain [P2S7]4− group, in which one [PS4]3− tetrahedron connects with its adjacent [PS4]3− tetrahedron through a bridging S anion. This is different from the traditional understanding of the presence of longer-chain [P2S7]4− covalent network would impeding rotational behavior of [PS4]3− tetrahedron and Li ion migration9,31. To further analyze the reason why [PS4]3− tetrahedra from longer-chain [P2S7]4− group have better rotation abilities than simple [PS4]3− group in Li7P3S11, P-S bonding strengths were quantified by COHP and ICOHP analysis. Viewed from Fig. S14, -ICOHP value of P8-S28 bond is smaller than those of P8-S4 and P3-S11 bonds, demonstrating bonding strength of P-S pair with a bridging S anion is weaker than those P-S pairs with terminal S anions. On the other hand, bond lengths of 2.14–2.17 Å for P8-S28 are larger than 2.00–2.05 Å for other P-S bonds without bridge S atom. The longer bond length means the weaker bonding strength, in good accordance with the results of COHP calculations. These weak electronic interactions between P8 cation and S28 anion make P8-S28 bond effectively act as the rotation axis for [PS4]3− tetrahedron.

Fig. 2: 1μs MLMD simulations for Li7P3S11.
figure 2

a Initial crystal structure before MLMD simulations, b final structures with PS4 polyanion rotations after 1μs MLMD simulations at 300 K, and c S4-S5-S44 in PS4 polyanion rotating along P8-S28 bond axis. Schematic diagrams of the definition of angles θ and φ for d [PS4]3− tetrahedron and e [ErCl6]3− octahedron in the reference coordinate system. θ is defined as the angle between P–S or Er-Cl bond and z-axis, while angle φ corresponds to angle between the y-axis and the projection of P–S or Er-Cl bond vector in the xy plane.

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Detailed analysis of atomic trajectories from 1μs MLMD simulations were performed to gain more comprehensive understanding of polyanion rotation in five lithium ionic conductors. To reduce computational expense for trajectory data analysis, here, crystal structure snapshots every 50 ps were extracted from 1μs MLMD simulations, and saved as atomic trajectory file. Figure S15 shows that P–S bond lengths in 5#, 6#, 7#, and 8# [PS4]3− tetrahedron are fluctuating around 2.05 Å due to atomic vibrations or rotations during the whole 1 μs simulation time, and P-S bonds are well maintained even upon [PS4]3− polyanion rotations, indicating that crystal structure is stable in dynamics at room temperature. To expediently monitor polyanion rotation during the whole MLMD simulation time, θ and φ angles were used as the indicators for [PS4]3− and [MX6]3− (M is transition metal, X is halogen) rotation, as depicted in Fig. 2d, e. Figure 3 shows the evolutions of θ and φ angles of four P-S bonds in 5#, 6#, 7# and 8# [PS4]3− polyanions in Li7P3S11 with respect to simulation time. It can be clearly seen that the remarkable exchanges in θ and φ angles occur at ~0.71 μs, ~0.71 μs, ~0.85 μs and ~0.92 μs for the first time, respectively for 5#, 6#, 7# and 8# [PS4]3− polyanions. It reveals substantive rotations of [PS4]3− polyanion in Li7P3S11 at 300 K. Because of the room-temperature [PS4]3− polyanion rotations usually occurring at the last two thirds of our 1 μs MLMD simulations, so it’ is no wonder that the traditional room-temperature AIMD simulation at 300 ps level could not capture polyanion rotation in Li7P3S1132. Figure 4 shows θ and φ angle evolutions of four P-S bonds in 7# and 8# [PS4]3− tetrahedra from 918.05 ns to 918.20 ns with a time interval of 0.5 ps. It can be seen that a half rotation of three S ligands in [PS4]3− polyanion over ~60 degrees occur within a very short time interval of ~3 ps, and a full rotation over 120 degrees for [PS4]3− polyanion was finished in ~6 ps (from 918.137 ns to 918.143 ns), as Fig. S17 and video shown in Supporting Information. According to Li-Li average distance (Fig. S18) and Li diffusion coefficient (Fig. 1c), the estimated time scale of ~10 ns for a complete Li hopping between two stable Li sites at room temperature is much larger than a full [PS4]3− rotation, indicating the correlation between [PS4]3− rotation and Li ion translational motion is improbable at least on a time scale.

Fig. 3: Structural characteristic of PS4 polyanion rotations during the whole 1μs MLMD simulations for Li7P3S11 at 300 K.
figure 3

Angle θ and φ evolutions of the four P-S bonds in a, b 5#, c, d 6#, e, f 7#, and g, h 8# [PS4]3− tetrahedra, respectively. Here, two adjacent data points with a time interval of 5 ps.

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Fig. 4: Angle evolutions of P-S bonds.
figure 4

θ and φ evolutions of the four P-S bonds in a–c 7# and b–d 8# [PS4]3− tetrahedra for Li7P3S11 during 300 K MLMD simulations from 918.05 ns to 918.20 ns. Here, two adjacent data points with a time interval of 0.5 ps.

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To quantitate the rotation dynamics, the Helmholtz free energies of four S anion ligands in [PS4]3− polyanion (Fig. 5) were calculated from 2D probability distribution ρ(θ,φ) by mapping four S ligands in the spherical coordinates defined in Fig. 2d. The Helmholtz free energy surface reflects rotation routes by the coordinates of θ and φ, and rotational activation energy barriers by height difference between the local minima and transition states. It is worth noting that a remarkable [PS4]3− rotation needs at least three S ligands simultaneously exchanging their spherical coordinates of θ and φ angles and staying at their adjacent minima, so at least three energy landscapes should be considered for each [PS4]3− polyanion. It can be seen from Fig. 5 that for each [PS4]3− polyanion, although one shallow energy landscape between two S ligands with a relatively low energy barrier (0.34 eV, 0.25 eV, 0.27 eV and 0.30 eV for 5#, 6#, 7# and 8# [PS4]3− polyanion, respectively) is noticed, the overall energy barriers for 5#, 6#, 7#, and 8# [PS4]3− polyanion rotational dynamics are in the range of 0.60–0.68 eV, which are much higher than those of Li diffusion in above lithium ionic conductors (Fig. 1), nonetheless close to Li diffusion in some common cathode materials, such as LiCoO2 (0.73 eV)33,34, LiFePO4 (0.60 eV)35,36, and LiMn2O4 (0.58 eV)37. In terms of such energy barrier magnitudes (0.60–0.68 eV), although the occurrence probability of [PS4]3− polyanion rotations in Li7P3S11 at room temperature is smaller than the translational motion of Li ion, rotating [PS4]3− polyanion by the thermal activation stills relatively easy to produce, consistent with remarkable rotations in Figs. 3, 4, and S17 and video in Supporting Information.

Fig. 5: Rotational dynamics of [PS4]3− polyanion.
figure 5

The Helmholtz free energy surface of P-S bonds as a function of θ and φ angle for a, b 5#, c, d 6#, e, f 7#, and g, h 8# [PS4]3− tetrahedron, respectively, from the whole 1μs MLMD simulations for Li7P3S11 at 300 K. The free energy F was computed as \(F(\theta ,\varphi )=-{k}_{B}{\rm{T}}{\rm{ln}}[\rho (\theta ,\varphi )]\), where kB is the Boltzmann constant, T is temperature, and ρ(θ, φ) is 2D projected probability distribution of four S ligands (as shown in Fig. S18 in Supporting Information).

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Furthermore, the elevated temperature MLMD simulations of 350 K beyond room temperature were performed to explore enhanced [PS4]3− polyanion rotations in Li7P3S11, and the corresponding total simulation times are as the same as those for 300 K. Figure S16 shows P-S bond lengths are fluctuating around 2.05 Å during the whole 1 μs simulation time, and all P-S bonds are well maintained even upon intense [PS4]3− polyanion rotations, demonstrating stable structure dynamics at elevated temperature. Viewed from Fig. S20, for each 5#, 6#, 7#, and 8# [PS4]3− tetrahedron, three S ligands continually exchange their spherical coordinates of θ and φ and stay at their adjacent minima during the whole 1 μs simulation time, indicating remarkable [PS4]3− rotations in Li7P3S11 at 350 K. Additionally, for each 5#, 6#, 7# and 8# [PS4]3− tetrahedron, Helmholtz free energy surfaces (Fig. S21) clearly show flat energy landscapes among three S ligands, corresponding to the relatively low energy barriers (0.16–0.21 eV) for intensive [PS4]3− rotations. Making a comparison between MLMD simulation results at 300 K and 350 K, [PS4]3− rotational event in Li7P3S11 is very sensitive to temperature change and thermal activation, and the occurrence frequency of rotation can be enhanced several times just by increasing 50 K. Therefore, extrapolating polyanion rotational events from elevated temperatures (>600 K) AIMD simulations to room temperature situation may cause some underlying deviations.

For other lithium conductors, 1μs MLMD simulations at 300 K were also performed, and the simulation results are shown in Figs. S22–S26. During the whole MLMD simulations time, no anion ligand exchanges its spherical coordinates of θ and φ, and no rotation at room temperature is observed for [PS4]3− and [GeS4]4− tetrahedron Li10GeP2S12, [PS4]3− tetrahedron in β-Li3PS4, [ErCl6]3− octahedron in Li3ErCl6, and [YBr6]3− octahedron in Li3YBr6, respectively. It should be noted that our MLMD simulations just demonstrate no remarkable polyanion rotation at room temperature during the 1μs simulation time, and polyanion rotations beyond 1μs cannot be completely ruled out for Li10GeP2S12, β-Li3PS4, Li3ErCl6, and Li3YBr6 at 300 K. But certainly, the probability of occurrence of polyanion rotation in Li10GeP2S12, β-Li3PS4, Li3ErCl6, and Li3YBr6 at room temperature is relatively much lower than Li7P3S11.

The weakly negative effect of polyanion rotation on Li ion migration

To answer the question of whether [PS4]3− polyanion rotation would effectively facilitate Li ion diffusion in Li7P3S11 at 300 K, we further explored the relationships between [PS4]3− polyanion rotation and Li ion translational motion. Figure 6a–d show 2D probability density distributions (\({\rho }_{r,\theta }^{2D}\)) of P-S-Li angles θ and distance r between S ligands and their adjacent Li ions for rotational 5#-, 6#-, 7#- and 8#-[PS4]3− polyanions. It is seen that \({\rho }_{r,\theta }^{2D}\) of four rotational [PS4]3− polyanion systems are quite delocalized and dispersed in the most space of (r, θ), signifying the quite weak polyanion-cation dynamical couplings. On the other hand, if there is a strong polyanion-cation coupling dynamics, \({\rho }_{r,\theta }^{2D}\) would be localized in an agminated spot. Li ions in Li7P3S11 are divided into two groups, that is Li ions in group-I are close to rotational [PS4]3− polyanions, and Li ions in group-II are far away from rotational polyanions. Figure 6e shows e diffusion trajectories of Li ion in group-I near rotational 5#- and 6#-[PS4]3− polyanions from 711.207 ns to 711.212 ns. During the rotation time of 5 ps, two [PS4]3− polyanions successfully rotate 120 degrees, and Li ions in group-I close to these two polyanions also dramatically hop rather than vibration at their equilibrium sites, whose hopping distances are more than 1 Å. A similar phenomenon is observed for Li ion diffusion around rotational 7#- and 8#-[PS4]3− polyanions from 918.137 ns to 918.143 ns, as depicted in Fig. 6g.

Fig. 6: The relationships between [PS4]3− polyanion groups (5#-P, 6#-P, 7#-P and 8#-P) and their adjacent Li ions during [PS4]3− rotation time for 300 K MLMD simulations of Li7P3S11.
figure 6

2D probability density distribution (\({\rho }_{r,\theta }^{2D}\)) of P-S-Li angles θ and distance r between S ligands and Li ion in the first shell of a 5#-, b 6#-, c 7#-, and d 8#-[PS4]3− polyanions, respectively; e diffusion trajectories of each Li ion near rotational 5#- and 6#-[PS4]3− polyanions; f Pearson correlation coefficients between angular velocity vectors of S ligands in rotational 5#- and 6#-[PS4]3− polyanions and their adjacent Li ions; g diffusion trajectories of each Li ion near rotational 7#- and 8#-[PS4]3− polyanions; h Pearson correlation coefficients between angular velocity vectors of S ligands in rotational 7#- and 8#-[PS4]3− polyanions and their adjacent Li ions; evolutions of diffusion coefficient for Li groups near and far away from rotational i 5#-, j 6#-, k 7#-, and l 8#-[PS4]3− polyanions, respectively.

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To quantitatively determine the correlations between [PS4]3− polyanion rotation and Li ion translational diffusion, the Pearson correlation coefficients38 between angular velocity vectors of S ligands in rotational 5#-, 6#-, 7#-, and 8#-[PS4]3− polyanions and their adjacent Li ions were calculated, as shown in Fig. 6f, h. Here, only those Li ions staying within the first shell of [PS4]3− during the whole rotation time are considered for the Pearson correlation coefficient calculations, whose P-Li distances are less than 4 Å, referred from RDF of P-Li in Fig. S18. All the absolute values of calculated Pearson correlation coefficient are less than 0.25. Given that a Pearson correlation coefficient smaller than 0.5 is generally regarded as weak in statistics39, so Li ion translational diffusion is quite weakly correlated to rotational [PS4]3− polyanions. Furthermore, diffusion coefficients of Li group-I near and Li group-II far away from rotational polyanions were determined to get detailed diffusion events, as shown in Fig. 6i–l. It is found that diffusion coefficients of Li group-II far away from [P2S7]4− are smaller than those of Li group-I near [P2S7]4−. As Fig. S27 shows that the calculated anion charges of S ligands in [P2S7]4− are smaller than those of isolated [PS4]3− groups, especially for the bridging S anion with a charge of −0.57 e, so the electrostatic attractions between [P2S7]4− polyanion group and Li+ cation are relatively weaker40,41,42, eventually leading to higher diffusion diffusivities.

On the other hand, most importantly, diffusion coefficients of both Li groups near and far away from rotational 5#-, 6#-, 7#-, and 8#-[PS4]3− polyanions unanimously exhibit downtrends within [PS4]3− rotation time. Because the Li-anion electrostatic interactions are long-ranged, rotational [PS4]3− polyanions also have impacts on those Li ions far away from them. In other words, the rotation of [PS4]3− polyanion groups would lower the instantaneous Li ion diffusivities of all ions in unit cell, which may be due to the contingent inconsistency of direction between Li ion translational motion and polyanion rotation disturbing the potential energy surface of Li with respect to [PS4]3− polyanion rotation. It is fully consistent with our previous results of anharmonic phonon calculations for crystalline Li7P3S1143, that is the low-frequency [PS4]3− rotational modes anharmonically couple with high-frequency Li modes, which vibrate toward S-S edge or along Li-S apex and are not Li diffusion pathways. The phenomenon of decoupled [PS4]3− rotational motion from Li jump is also observed in 70Li2S-30P2S5 glass system32. Therefore, from our MLMD simulations at 300 K, it can be concluded here that there is no paddle-wheel effect in the crystalline Li7P3S11 at room temperature, and rotational [PS4]3− polyanion groups at 5–6 ps level have some weakly negative impacts on the overall Li ion diffusion in Li7P3S11 unit cell. In fact, lack of paddle-wheel effect is also reported for BH4-substituted Li argyrodite system44. Compare to other sulfides (Table S1), the superionic conductivity of for Li7P3S11 mainly owes to its highly distorted LiSn polyhedron and the weak Li-anion Coulomb force providing the frustrated potential energy surface for Li ion migration with low activation energy barrier4. Li ion migration in Li7P3S11 benefit from bridging S anions from longer-chain [P2S7]4− units rather than rotations of [PS4]3− polyanion from them.

In this work, the machine-learning interatomic potentials based on the MTP framework were developed for Li7P3S11, Li10GeP2S12, β-Li3PS4, Li3ErCl6, and Li3YBr6 lithium ionic conductors. The newly developed machine-learning interatomic potentials well reproduce the DFT energies and forces, lattice parameters, and local structures of these five lithium ionic conductors. Combining the classical molecular dynamics simulations with our developed machine-learning interatomic potentials, the ultralong-time MLMD simulations were performed for five lithium ionic conductors at near room temperature and other elevated temperatures, which enhance the sampling of Li diffusion events and reduce calculation errors. On one hand, MLMD simulations show Li7P3S11, Li10GeP2S12, β-Li3PS4, and Li3YBr6 exhibit the linear Arrhenius behaviors from 300 K to 500 K. While for Li3ErCl6, the non-linear Arrhenius behavior was observed, and the corresponding phase transition occurs at ∼460 K. The near room temperature MLMD simulations well capture the non-linear Arrhenius behavior and would increase the accuracy of calculated ionic conductivity at room temperature. On the other hand, the simulation time of 300 K MLMD simulations was extended to 1 μs level to capture sufficient Li diffusion and polyanion rotation events in lithium ionic conductors at room temperature. MLMD simulation results demonstrate no remarkable polyanion rotation in Li10GeP2S12, β-Li3PS4, Li3ErCl6, and Li3YBr6 at room temperature during 1μs simulation time, just librational motion for them. However, in Li7P3S11 at room temperature, four [PS4]3− tetrahedron in unit cell are noticed with remarkable rotational motions. All the rotated [PS4]3− tetrahedra belong to a part of the longer-chain [P2S7]4− group, while isolated group [PS4]3− does not rotate. Moreover, a full rotation of three S ligands in [PS4]3− polyanion over ~120 degrees was finished in 5–6 ps. Lastly, our 300 K MLMD simulations show Li ion translational diffusion is quite weakly correlated to the rotation of [PS4]3− polyanions in Li7P3S11, and rotational [PS4]3− polyanion groups at 5–6 ps level have some weakly negative impacts on the overall Li ion diffusion. There is no paddle-wheel effect in the crystalline Li7P3S11 at room temperature, and Li ion migration in Li7P3S11 benefits from bridging S anions from longer-chain [P2S7]4− units rather than rotations of [PS4]3− polyanion from them. In short, our ultralong-time MLMD simulations at low temperatures effectively capture the polyanion rotation events and identify no paddle-wheel effect in Li7P3S11 at room temperature, deepening our understanding of the relationship between polyanion rotation and cation diffusion in ionic conductors at room environment.

Methods

Crystal structures

Supercells of Li7P3S11, Li10GeP2S12, β-Li3PS4, Li3ErCl6 and Li3YBr6 with lattice parameters greater than 10 Å were constructed to minimize interactions between two periodic images. For Li7P3S11, a 1 × 2 × 1 supercell with 84 atoms was constructed from the experimentally determined crystal structure by Yamane et al.45. For Li10GeP2S12, a 2 × 2 × 1 supercell with 200 atoms was created from the computed structure in Materials Project (MP) database (mp-696138). For Li3ErCl6 and Li3YBr6, starting from the experimentally reported disordered structures with fractional occupancies, the 1 × 1 × 2 and 2 × 1 × 1 supercells with 60 and 80 atoms were firstly built, respectively, and then all Li-vacancy cation configurations were created using the enumeration method46 implemented in Pymatgen code47. Lastly, the most stable Li-vacancy configuration of Li3ErCl6 and Li3YBr6 were determined by ranking their DFT energies, and the corresponding space groups are P321 and C2/c, respectively.

DFT calculations and AIMD simulations

All calculations were carried out in the framework of density functional theory (DFT)48 using the projector augmented wave method49, as implemented in Vienna ab-initio Simulation Package (VASP). The generalized gradient approximation (GGA)50 and Perdew–Burke–Ernzerhof (PBE) exchange functional were employed. The plane-wave energy cutoff was set to 520 eV, and a k-point density of at least 64/Å−3, similar to those used in the MP database. AIMD calculations were performed without spin-polarization in an NPT ensemble at elevated temperatures with a Nose−Hoover thermostat51. A smaller plane wave energy cut-off of 300 eV was chosen for AIMD simulations of the supercells with a Gamma-centered 1 × 1 × 1 k-point grid, and the simulation supercell sizes were at least 10 Å along each lattice direction. Time step was set to 2 fs, and all supercell systems were simulated for a short time with a total of 10,000 steps. AIMD simulations with supercell model were performed at six different temperatures (from 300 to 1300 K with 200 K intervals) to diversify the training structures. For each temperature, about 300 snapshot structures were uniformly extracted from the production run of 20 ps. Hence, for each lithium ionic conductor, a total of 1800 training structures (300 × 6 temperatures) were generated. To obtain accurate energies and forces for machine learning potential training, the static self-consistent field (SCF) calculations for training structures were carried out. These SCF calculations were performed with a higher k-point density of at least 100/Å−3, an energy cutoff of 520 eV. The convergence criterion of energy for SCF calculations was set to 10−6 eV atom−1. The crystal orbital Hamilton population (COHP) between neighboring oxygen ions was computed by Lobster program52, in which the negative and positive COHP values indicate bonding and anti-bonding, respectively. Integrating COHP to Fermi energy level obtain -ICOHP data to quantify the chemical bonding strength. The pbeVaspFit2015 basis sets were used for the reconstruction of the PAW wave functions of each element.

MTP model training and verification

Today, there are many machine-learning atomic potential models, such as the neural network potential (NNP)53, the Gaussian approximation potential (GAP)54, the moment tensor potentials (MTP)55, the smooth overlap of atomic positions (SOAP)56 and the spectral neighbor analysis potential (SNAP)57. According to the previous work by S.P. Ong et al.25, MTP model exhibits the smallest root-mean-square errors in energies and forces. MD simulations based on the MTP framework have been successfully applied to many battery materials, including metals58, cathode coating materials26 and SSE materials27. We sure that MTP model is the best model for our work, when considering a trade-off between accuracy and computational cost. In this work, the energy and force data points are assigned weights of 100:1, similar to the previous work by Ong27. In the MTP framework, two key parameters should be carefully selected, which determine the potential performance, including radius cutoff (Rcut) and maximum level (levmax). The radius cutoff Rcut controls the maximum interaction range between atoms, in this work, Rcut for our five lithium ionic conductors are set to 5 Å, a typical value used in previously reported MTP frameworks26,27,58. The maximum level (levmax) determines completeness of the basis set functions, the larger levmax would make the larger number of terms in the linear expansion, in turn resulting in higher computational loading and a greater possibility of over-fitting. In this work, levmax for our five lithium ionic conductors were tested among these five values of 8, 10, 12, 14, 16 and 18. For the potential fitting, an 80:20 split of the training : test data was used. All training, evaluations and simulations with MTP framework were performed by using the open-source software and Python packages, including MLIP59, LAMMPS60 and Materials Machine Learning (maml).

Diffusivity and activation barrier calculations

Classical MD simulations for five lithium ionic conductors were performed by LAMMPS interfaced with the trained MTPs, called machine learning driven molecular dynamics (MLMD). The supercells of five lithium ionic conductors for MLMD simulations are the same as those of AIMD simulations. Time step was set to 1 fs, a typical value used in the previous work61. Each lithium ionic conductor was firstly equilibrated in the NPT ensemble for 100 ps, and then equilibrated in the NVT ensemble for 1 million steps (total simulation time as long as 1 ns). Lattice volumes for MLMD simulations in the NVT ensemble were averaged from all structures of the previous MLMD simulations in the NPT ensemble. For each material, MLMD simulations were performed at room and slightly elevated temperatures (from 300 to 520 K). At each temperature, MLMD simulations were repeated 20 times to enhance samplings of diffusion events and reduce the calculation errors.

Diffusional properties of an atom can be calculated from its MLMD trajectories with positions of ri(t), and the displacement Δri of i-th particle from time t1 to t2 can be calculated by \(\Delta {r}_{i}(\Delta t)={r}_{i}\left({t}_{2}\right)-{r}_{i}\left({t}_{1}\right)\), where \(\Delta t={t}_{2}-{t}_{1}\). Total squared displacements are obtained by summing the squared displacements of all N mobile atoms over a time interval Δt, \({\sum }_{i=1}^{N}({|{\Delta r}_{i}(\Delta t)|}^{2})\). Over a total MLMD simulation time duration of ttotal, there are many time intervals of NΔt with the same duration of Δt (Δt < ttotal) at different starting time of t. Due to atom’s displacements over Δt reflecting mobility of atoms, the total mean squared displacements (TMSDs) of all N mobile atoms over a time interval of Δt can be obtained by calculating the statistical ensemble average of the squared displacements over a total of NΔt time intervals with the same duration of Δt62, \({\rm{TMSD}}(\varDelta t)={\sum }_{i=1}^{N}\langle {|{r}_{i}(\varDelta t)-{r}_{i}(0)|}^{2}\rangle ={\sum }_{N}^{N}\frac{1}{{N}_{\varDelta t}}{\sum }_{t=0}^{{t}_{total-\varDelta t}}{|{r}_{i}(t+\varDelta t)-{r}_{i}(t)|}^{2}\), where \(\left\langle \cdot \right\rangle\) stands for the statistical ensemble average. This statistical ensemble average over different time intervals of NΔt offers the statistical analysis of sufficient diffusional events to obtain accurate diffusional properties. To get diffusivity of the mobile species, MSD over time interval of Δt is calculated by averaging TMSDs to each mobile atom, MSD(Δt) = TMSDs(Δt)/N, where, N is the number of mobile atoms.

If MLMD simulations contain the sufficient diffusional events, the dependence of MSD over time intervals of Δt would follow a linear relationship. According to the Einstein relation, self-diffusion coefficient D of a specie can be calculated from the slope of MSD curve as a function of time intervals Δt 63, \(D=\frac{{\rm{MSD}}(\varDelta t)}{2d\varDelta t}\), where, d = 3 is the diffusion dimension of a particle in simulated system. The linear fitting of MSD vs. Δt to the Einstein relation should only be performed on the linear region corresponding to good diffusional displacements. Therefore, to achieve small error bounds of the fitted D and get accurate diffusivity, MLMD simulations should be long enough for capturing a large number of diffusion events. Performing a series of MLMD simulations at different temperatures obtains the Arrhenius relations of the log of diffusivity D as a function of 1/T, \(D={D}_{0}{\rm{\exp }}\left(-{E}_{{\rm{a}}}/{k}_{B}T\right)\), and which can extrapolate the prefactor D0, overall activation energy Ea, diffusivity D and conductivity σ at RT.