Lithium Ion Transport Environment by Molecular Vibrations in Ion‐Conducting Glasses

Controlling Li ion transport in glasses at atomic and molecular levels is key to realizing all‐solid‐state batteries, a promising technology for electric vehicles. In this context, Li3PS4 glass, a promising solid electrolyte candidate, exhibits dynamic coupling between the Li+ cation mobility and the PS43− anion libration, which is commonly referred to as the paddlewheel effect. In addition, it exhibits a concerted cation diffusion effect (i.e., a cation–cation interaction), which is regarded as the essence of high Li ion transport. However, the correlation between the Li+ ions within the glass structure can only be vaguely determined, due to the limited experimental information that can be obtained. Here, this study reports that the Li ions present in glasses can be classified by evaluating their valence oscillations via Bader analysis to topologically analyze the chemical bonds. It is found that three types of Li ions are present in Li3PS4 glass, and that the more mobile Li ions (i.e., the Li3‐type ions) exhibit a characteristic correlation at relatively long distances of 4.0–5.0 Å. Furthermore, reverse Monte Carlo simulations combined with deep learning potentials that reproduce X‐ray, neutron, and electron diffraction pair distribution functions showed an increase in the number of Li3‐type ions for partially crystallized glass structures with improved Li ion transport properties. Our results show order within the disorder of the Li ion distribution in the glass by a topological analysis of their valences. Thus, considering the molecular vibrations in the glass during the evaluation of the Li ion valences is expected to lead to the development of new solid electrolytes.


Introduction
Li-ion batteries are widely used power sources in mobile devices and power tools.They are also used in plug-in hybrid electric vehicles and electric vehicles (EVs), [1] and their use is expected to increase because of their ability to significantly reduce CO 2 emissions. [2]With regard to battery safety, there is a strong demand to replace the current flammable organic electrolyte with a non-flammable and flame-resistant solid electrolyte to produce allsolid-state batteries.Such developments would ensure the safety of large batteries for EVs, and further developments would be aimed at further extending the mileage of such vehicles.
Examples of solid electrolytes include oxides, [3][4][5] polymers, [6,7] and sulfide-based electrolytes, [8] with the sulfide-based electrolytes being the closest to current commercial products in the large battery market because of their high Li ion conductivity and their ease of battery fabrication.More specifically, all-solidstate Li-ion secondary batteries based on flameresistant solid electrolytes prepared from Controlling Li ion transport in glasses at atomic and molecular levels is key to realizing all-solid-state batteries, a promising technology for electric vehicles.In this context, Li 3 PS 4 glass, a promising solid electrolyte candidate, exhibits dynamic coupling between the Li + cation mobility and the PS 4 3− anion libration, which is commonly referred to as the paddlewheel effect.In addition, it exhibits a concerted cation diffusion effect (i.e., a cation-cation interaction), which is regarded as the essence of high Li ion transport.However, the correlation between the Li + ions within the glass structure can only be vaguely determined, due to the limited experimental information that can be obtained.Here, this study reports that the Li ions present in glasses can be classified by evaluating their valence oscillations via Bader analysis to topologically analyze the chemical bonds.It is found that three types of Li ions are present in Li 3 PS 4 glass, and that the more mobile Li ions (i.e., the Li3-type ions) exhibit a characteristic correlation at relatively long distances of 4.0-5.0Å.Furthermore, reverse Monte Carlo simulations combined with deep learning potentials that reproduce X-ray, neutron, and electron diffraction pair distribution functions showed an increase in the number of Li3-type ions for partially crystallized glass structures with improved Li ion transport properties.Our results show order within the disorder of the Li ion distribution in the glass by a topological analysis of their valences.Thus, considering the molecular vibrations in the glass during the evaluation of the Li ion valences is expected to lead to the development of new solid electrolytes.
materials such as lithium sulfide (Li 2 S) have been shown to have superior safety profiles to conventional batteries.In addition, because the electrode layers are composites of an electrode active material and a solid electrolyte, the energy density can be improved by increasing the amount of the active material. [9]Therefore, the corresponding solid electrolyte should possess a high Li ion conductivity; sulfide solid electrolytes based on Li 2 S are applicable for such systems since their ion conductivities are comparable to those of organic electrolytes. [10]The development and design of advanced solid electrolytes therefore relies heavily on our understanding of the correlation between the electrolyte structure and the Li ion conductivity.For example, for achieving superionic behavior using the most conventional solid electrolytes for solid-state batteries, lithium and sodium thiophosphates [11][12][13] have been designed based on the framework structure, [14] lattice polarizability, [15] activation energy factor, [16] and cation-vacancy correlation. [17]However, these experimental and computational studies have focused on the relationship between the static structure and the ionic conduction properties.
In contrast, ab initio molecular dynamics (AIMD) simulations have shown that in lithium argyrodites, i.e., Li 6 PS 5 X (X = Cl, Br, I), disordered substitutional anions enable the fast transport of Li ions. [18]For example, Adelstein et al. showed that fluctuations in the chemical environment due to the presence of polarizable bromide anions in Li 3 InBr 6 contribute to its superionic behavior. [19]For sodium thiophosphate, a combination of the neutron diffraction maximum-entropy method and AIMD simulations showed that anion rotation significantly enhances the transport of Na ions in Na 11 Sn 2 P n X 12 (Pn = P, Sb; X = S, Se). [20]Similarly, Sigel et al. used AIMD simulations for Li 3 PS 4 glass to explain the paddlewheel effect at low temperatures in the glass structure. [21]The paddlewheel effect refers to the dynamic coupling of Li + migration to the libration motion of the PS 4 3− tetrahedra.They found that the rotation of the PS 4 complex anions present in the glass exerts a force on the cations.In addition, the glass is less dense than its crystalline analog, γ-Li 3 PS 4 , owing to incomplete ion packing (i.e., an amorphous structure), and this lower density provides the additional free volume required to enable anion rotation.Furthermore, while a low density (i.e., compared to the crystalline phase) is a common property of all glasses, the Li 3 PS 4 composition is characterized by the relative absence of a covalent bonding network consisting of long-chain P x S y anions.In support of these studies, it was also proposed that the underlying rotational motion effect of polyanion, which is actually inhibited by the substitution of larger polyanions in single-phase solid solution region, causes the unexpected lowering of the ionic conductivity. [22]o understand the relationship between the anion framework structure and the dynamics (self-diffusion) of Li ions, quasielastic neutron scattering (QENS) was used to examine Li 7 P 3 S 11 glass ceramics. [23]The observed QENS component can be regarded as the self-diffusion of Li ions, as in the following equation: where Γ is the half-width at half-maximum of the component, and τ 0 is the mean residence time of the Li ions.Γ can be decomposed from the jump-diffusion model using Q 2 as follows: where Q is the magnitude of the scattering vector, D s is the selfdiffusion constant, and <l> is the mean jump length.Using these equations, the self-diffusion coefficient for the Li + ions (D S_Li ) was 5.7 × 10 −6 cm 2 s −1 , the mean residence time τ 0 was 5.5 × 10 −11 s, and the mean jump length <l> was 4.3 Å.In addition, the ionic conductivity of this system was 1.7 × 10 −2 S cm −1 . [9]nterestingly, Matsuura et al. evaluated the D S_Li of the oxide solid electrolyte La 2/3 − x Li 3x TiO 3 (LLTO) in a QENS experiment [24] and found that it was 3.1 × 10 −6 cm 2 s −1 , thereby indicating that the D S_Li values of LPS and LLTO were comparable.In addition to this experimental evaluation of the Li jump length on the microscopic scale, they evaluated the fraction of moving Li ions by comparing the QENS intensity to the elastic scattering intensity and found the same fractions for LPS and LLTO.Furthermore, they stated that the difference in the macroscopic Li ion conductivity can be attributed to the Li density in the unit cell.In contrast, using the 7 Li pulsed field-gradient (PFG) NMR technique, the value of D S_Li was determined to be 9.0 × 10 −10 cm 2 s −1 in β-Li 3 PS 4 , [25] and its macroscopic Li ion conductivity was estimated to be 1.0 × 10 −4 S cm −1 , which is consistent with that obtained experimentally from the impedance results (i.e., 2.0 × 10 −4 S cm −1 , see Table 1).[28][29] Hence, finding the key parameters that lead to improvements in D S_Li and in the Li distribution will be expected lead to more ionic conductivity in glassy electrolytes.Thus, we aim to determine the number of Li ion types that are present in Li 3 PS 4 glass, in addition to evaluating their characteristic correlations at various distances by examining their valence oscillations.For this purpose, Bader analysis was employed to topologically analyze the chemical bonds present in the glass.Furthermore, we aim to develop an understanding of the correlation between the Li ion distribution and Li ion transport in partially crystallized sulfide-glass structures with improved Li ion transport properties by reverse Monte Carlo simulations combined with deep learning potentials that reproduce the pair distribution functions obtained from X-ray and neutron diffraction experiments.1a, which shows a structural model of the PS 43− anion for Li 3 PS 4 glass at 300 K and for the eight Li ions that are ionically bonded to the anion, one anion from the AIMD simulation was taken as an example.During the 1.6 ps simulation employed herein, the model plots the ion positions every 0.2 ps.In the solid form (i.e., crystals or glasses), the constituent molecules and ions undergo constant thermal vibration. [30]As can be seen in Figure 1a, the anions and cations move significantly within the glass, even during the short 1.6 ps time period.In addition, Figure 1b, c show the trajectories of an S-ion libration and a Li ion vibration every 0.002 ps, and the change in each axis (i.e., the x-, y-, z-axes) from the initial coordinates is shown.Compared with the vibration of the Li ion, the migration of the S ion is a gradual shift owing to the sp 3 hybridization of the PS 4 3− anion (See Figure S1, Supporting Information, for the mean square displacement of each ion).We observed anion libration, such as the paddlewheel effect, as reported by Smith et al., [21] and confirmed that the Li ions were significantly affected by the S-ion libration.Since a total of 800 simulation time series structural models were generated every 0.002 ps during the 1.6 ps simulation, the time-averaged structure factors could be calculated, which were found to reproduce the experimental diffraction data [14] extremely well, as shown in Figure S2, Supporting Information.These results indicate the importance of considering the molecular vibration when discussing the correlation between the structure and the degree of Li ion conduction in a glassy structure where simple molecules form the framework structure. [21,31,32]n this context, Zhang et al. examined the dynamic response of the anion framework by joint-time correlation analysis, and they examined similar molecular vibrations of the PX 4 3− (X = S, Se) anion in Na 11 Sn 2 P n X 12 .They argued that cation mobility is dictated by the static framework structure, and that the anion dynamics affect the energy landscape perceived by the cations because anion libration and the cation mobility are strongly correlated. [20]Indeed, they clearly demonstrated that the velocities of cations and polyanions are synchronized (i.e., they are dynamically coupled).They also reported that a harmonious mechanism involving the paddlewheel effect (i.e., anion-cation interactions) and correlated diffusion (i.e., cation-cation interactions) underlies the intrinsically high ionic conductivities of such glasses. [20]Similarly, Smith et al. reported the existence of the paddlewheel effect at low temperatures (i.e., room temperature) for Li 3 PS 4 glass and noted that the Li-Li correlations are rather vague.The structural models of the ab initio MD time step also suggest that the Li ions possess various solvation environments, with coordination numbers ranging from 3 to 5.These observations suggest that Li migration occurs via a complex mechanism that combines the concerted motion of Li ions with the large semipermanent libration of the PS 4 3− tetrahedron.

Charge Vibrations of the Li Ions as Evaluated by Bader Charge Analysis
To clarify the correlation between the Li ions in detail, we attempted to estimate the positive charge of the Li ions in the glass structure using Bader charge analysis. [33]This analytical technique can estimate the charge of a Li ion from a structural model for which the charge density has been calculated because the charge for each ion is integrated with respect to the charge gradient.Thus, Figure 2a shows the crystal structure of the β-phase [34] (blue: P, green: S, and red: Li) and the charges contributing to the Li ions (yellow polyhedra).The valence of the Li ion can also be estimated from the charge of the polyhedron (Li ion valence = 1 − Bader charge).From the Bader charge analysis, we confirmed that the valence of the Li ion oscillated during the AIMD period of 1.6 ps, as shown in Figure 2b.Here, the mean of the valence during Energy Environ.Mater.2024, 7, e12612 the 1.6 ps simulation can be calculated from V i , and its var- The results of the calculations during the 1.6 ps period for the crystalline β-phase at 300 K, the γ-phase at 300 K, the γ-phase at 600 K, and the glass at 300 K are shown in Figure 2c.Although the mean valence of the Li ions is similar for crystals and glasses, their variances are larger in glasses than in crystals.This is reasonable, considering that the valence of the Li ions does not fluctuate significantly when the anionic framework structure is crystalline in nature.We therefore confirmed that even in the crystal form, the variance value increased with increasing temperature.More specifically, the distance between the position of the maximum charge in the charge cloud of each Li ion and the reapproaching Li ion position (see Figure 2d) was confirmed, as shown in Figure 2e.For this distance, the time averages and the variances during the 1.6 ps simulation were cal- , respectively, as in the case as for the valence calculations, and the results are shown in Figure 2f.The valence variance shown in Figure 2c is clearly different for the glass and the crystal; however, in terms of the distance variance, comparable values were obtained.Interestingly, the distance showed a distribution between 0.02 and 0.09 Å, with a minimum variance at ~0.055 Å (Figure 2d).This characteristic charge distribution is not significantly different between crystals and glasses, indicating that the charge distributions related to the Li ions are almost identical.Here, the distances with a small variance (i.e., from 0.045 to 0.065 Å) and large variances (i.e., <0.045 Å and >0.065 Å) are classified as Li2-type, Li1-type, and Li3-type, respectively.The obtained variance values indicate that the Li1-type and Li3-type systems are "unstable," while the Li2-type system is "stable."As shown in the charge clouds presented in Figure 3a, the distance between the charge maximum and the position of the Li ion is relatively constant for the Li1-type system, and so the charge cloud covers the Li ion almost perfectly.For the Li2-type system, the larger distance results in the charge cloud partially disappearing when the Li ion is centered; the Li3-type system shows an even stronger tendency, as can be seen upon examination of Figure 3a-c.As a result, the "unstable" Li1-type tends to move away from the center of charge, while the "unstable" Li3-type tends to move toward the center of charge.Curiously, AIMD simulations suggest that the charge cloud states, such as that of the Li2-type system, are stable.However, we can also consider that a variance of 0.016 at l = 0.02 Å for the Li1-type system indicates a change of ~0.04 Å from the mean value l h i, as described in the above equation, thereby suggesting that the Li1type system may have moved to the Li2type position at l = 0.06 Å.When a Li ion in the Li1-type state begins to move, it is easy to imagine a Li2-type charge distribution regardless of the direction in which it moves.Thus, it is easier to adopt the Li2-type state when moving, and it is conceivable that Li ions in the Li3-type state are more mobile than their Li1-type and Li2-type equivalents.This result therefore confirms that Li ions with a stable Li2-type charge were dominant (Figure 3d).This Li ion environment was also confirmed in AIMD simulations of lithium argyrodite Li 6 PS 5 Cl (see Figure S3, Supporting Information).

Evaluation of Li Ion Dynamics Using the Van Hove Function in AIMD Simulations
Based on the three Li classifications and the van Hove function, we analyzed the time-space dependence of the Li-Li correlations.Prior to the van Hove function analysis, the correlation between the Li classification and the Li coordination number for each Li ion was checked, and no specific correlation was found in the classified Li ions, as shown in Figure S4, Supporting Information.However, a clear feature can be discerned from these results.More specifically, as shown in Figure S4g-i, Supporting Information, which presents the coordination numbers of the Li1-type, Li2-type, and Li3-type species around the Li3-type ion versus the Li-Li ion distance, a plateau region is present at ~4.3 Å for the Li1type and Li2-type states around the Li3-type species in the βand γ-Li 3 PS 4 crystals.This distance also corresponds to a jump length of 4.3 Å for the Li ions reported in the QENS measurements. [23]However, no such correlation was found with the vacancies around the Li ions, suggesting that the Li ion conduction mechanism in the glass does not involve ionic conduction via vacancies, but instead takes place through the concerted motion of Li ions that conduct one another through correlation.In other words, the possibility of correlated diffusion (i.e., cationcation interactions) can be proposed.No such characteristic correlation was observed in the glassy structure, which is consistent with the discussion of Smith et al. [21] In this case, to perform local dynamics analysis in the glasses, focus should be placed on the correlations in space and time in systems of interacting particles, as introduced by van Hove. [35,36]Thus, the Li ions in the 1.6 ps AIMD simulation were each classified by Bader charge analysis, and the van Hove time-space correlation functions were checked for each Li classification, as shown in Figure 4a-c.The Li2-type species, which accounts for the majority of the Li ions, gradually increases in terms of the number of correlations beyond a distance of 3.0 Å.On the contrary, the Li1-type species possessed stable correlations with one another at approximately 3.5-4.0and 6.0-8.0Å, while the Li3-type centers exhibited stable correlations at approximately 4.0-5.0,6.0, and 8.0 Å.The van Hove correlations were then integrated during 1.6 ps (see Figure 4d), and as expected, there appeared to be different correlations for the Li3-and Li1-type states, with the Li2-type system being considered as an intermediate between the two.This indicates that in the backbone structure of the PS 4 3− anions, Li ions with disordered charge clouds, such as the Li3-type ions, correlate with one another over longer distances than those with charge clouds, such as the Li1-type species.Furthermore, this feature remained even over long distances, suggesting that the Li ion pathways may be networked within the glassy structure.Indeed, these observations are consistent with the QENS results, where an equivalent jump distance of 4.3 Å [23] was considered to represent the concerted motion of the Li3-type ions.Similar to the van Hove time-space correlation function for the glass, those for the βand γ-Li 3 PS 4 crystals were also confirmed, as shown in Figures S5, S6, S7, Supporting Information.In addition, a comparison of the Li3-Li3 van Hove correlation is shown in Figure 4e.Interestingly, although these correlations were found in the β-Li 3 PS 4 crystals, no such correlations were detected in the γ-Li 3 PS 4 crystals at 4.0-5.0Å, which is a characteristic result.Table 1 summarizes the ionic conductivities of these phases, and the presented data show that the presence of Li3-Li3 correlations at a distance of 4.0-5.0Å corresponds to high ionic conductivity, thereby indicating the possible origin of the Li ion conduction mechanism.

Discussion
By means of a real-space charge topology analysis [33,37,38] using the Bader charge analysis method, we found that the number of Li ions present in a glassy structure in the mobile state can be determined.A schematic of the charge clouds presented in Figure 3ac is shown in Figure 5. Interestingly, AIMD simulations show that the oscillation of Li ions, such as the Li2-type ions, is stable, whereas the Li1-type and Li3-type states are extremely unstable.We also found that the ionic conductivity may be higher when the backbone structure of the PS 4 3− anions is formed with charge clouds such as the Li3 type, and when it is correlated over relatively long distances (i.e., 4.0-5.0Å).This analysis based on Bader charge analysis extracts the maximum amount of information, in contrast to methods that show only two-body correlations, such as pair distribution functions (PDFs).However, such analysis is not possible without the availability of appropriate three-dimensional glass atomic configurations that also consider X-ray and neutron diffraction data, in addition to the electronic states.Therefore, to analyze the classification and distribution of Li ions in partially crystallized sulfide glasses with a high Li ion conductivity using the above analytical technique, it is extremely important to construct a suitable glassy structure model that reproduces the experimental PDF data.Thus, in the current study, we developed a method for this purpose.More specifically, based on the reverse Monte Carlo (RMC) method for the construction of such glass models, [39,40] we incorporated machine learning methods to rationally construct a structural model of glass by eliminating the arbitrariness in the electronic states and the atomic arrangements.Specific details can be found in the Supporting Information.In addition, AIMD simulations were performed in the Vienna Ab initio simulation package (VASP) using density and temperature as the parameters.Various structures were created, and the energies, E, of those structures were evaluated.Subsequently, the potentials were created by deep learning of these E values using the distance and angle information of the bonds present throughout the structures.Using this method, we could construct a glassy structure with approximately 3000 particles, which satisfied the experimental data and was energetically stable.Importantly, we succeeded in constructing Li 3 PS 4 glassy structures with different ionic conductivities.Figure 6a shows the X-ray structure factors S(Q), of the glassy phase, the crystalline γ-phase, and the partially crystallized glassy Li 3 PS 4 , while the corresponding densities and ionic conductivities are listed in Table 1.These results demonstrated that the ionic conductivity of the glass was higher than that of the crystal.In addition, in the glass, the ionic conductivity increased from 6 × 10 −4 to 8 × 10 −4 S cm −1 when the 510 rpm rotation time was increased from 116 to 130 h in the mechanochemical process and the glassy Li 3 PS 4 was partially crystallized.The precipitated crystalline phase was the γ-phase, as is evident from the overlapping Bragg peaks shown in Figure 6a.Furthermore, Figure 6b shows the reduced PDF obtained by Fourier transform of each structure factor using the following equation: The formation of the nanocrystalline phase was verified by transmission electron microscopy (TEM) and electron diffraction measurements.More specifically, Figure 6c shows a dark-field (DF) image of the sulfide glassy electrolytes after mechanochemical process for 130 h.The nanocrystallites exhibiting bright areas were mostly 20-30 nm in size, as indicated by the arrows.In addition, the electron diffraction pattern obtained from the DF image is shown in Figure 6d, wherein spots can be observed in the diffraction pattern.We therefore considered that the existence of such nanocrystalline phases, which are a minority component in partially crystallized glasses, will lead to an increase in the ionic conductivity compared to that of a pure sulfide glass.As such, we assumed this specimen to be a mixture of a glass and crystals, as outlined in the following equation.Subsequently, the percentage of the γ-phase was estimated from the differential PDF analysis using G(r) to give a value of 20.6%, as shown in Figure S8, Supporting Information.
As shown in Figure S8a, Supporting Information, the difference between the crystalline phase precipitated in the partially crystallized glass and the βand γ-phases was confirmed; the precipitated crystalline phase was considered to be the γ-phase.Furthermore, PDF analysis of only the extracted crystalline phase (Figure S8b, Supporting Information) showed a good agreement with the γ-phase over a wide r range.Because the ionic conductivity of the γ-phase is low, the remaining glass and interface phases are believed to be responsible for enhancing the ionic conductivity.We therefore removed the crystalline components and extracted the structure factors of the glass and interface phases (blue line in Figure 6a).The glassy phase, including the extracted interface phase and the glassy phase (dashed lines in Figure 6a,b), also showed slight differences.Thus, a suitable glassy structural model considering the electronic state was constructed for the glass and the glass containing the interface phase.This was achieved using the RMC method with deep learning (DL) potentials (Figure 7a), and the obtained structural model was also found to successfully reproduce the experimentally obtained structure factor S(Q), as shown in Figure 7b.Bader charge analysis was then applied to the constructed  three-dimensional structural model, and the Li classification of the RMC-DL models showed a tendency for the Li1-and Li2-type classifications to decrease and the Li3-type classification to increase when partially crystallized and in the case of an improved ionic conductivity, as shown in Figure 7c.The 4.0-5.0Å correlations of the Li3-type species are clearly connected by red lines, as shown in Figure 7d.
Although we believe that the increase in the Li3-type classification at 4.0-5.0Å is responsible for the improved Li ion conduction, it is necessary to consider why such a characteristic charge cloud arises from the geometrical arrangement in the first place.For example, the potential for a change in the anion coordination around the Li ions should be considered, since this could alter the charge clouds.To investigate this further, Figure S9a,b, Supporting Information, show the results of the angular distribution of the three-body correlations with S ions within 3 Å, taking the Li ion of each classification in the middle.As can be seen, there was no difference in the angular distribution, and the anions around the Li ions were the same in all classifications.Furthermore, to obtain the coordination environment of the Li + ions in detail, Voronoi polyhedron statistics were calculated by Voronoi tessellation analysis, [41,42] in which each ion was assigned a Voronoi index, i.e., <n 3 , n 4 , n 5 , n 6 >, where n i denotes the number of i-edged faces and Σ i n i is the total coordination number.The results for the Li-cantered Voronoi polyhedra calculated up to 2.3 Å are shown in Figure S9c, Supporting Information, and these results do not depend on the classification of the Li ions, which is consistent with the bond angle distribution.Thus, no clear difference in the PDF two-body correlations was observed among the three-body correlations.However, Bader's topological analysis revealed that the difference was due solely to charge, and not to changes in the interatomic distance.Although S ions are known to be highly polarized, this polarization does not alter the interatomic distance, thereby suggesting that charge exchange beyond the three-body correlation is responsible for the conduction of Li ions.In fact, the Li1-and Li3-type species have different van Hove functions, as shown in Figure 4, and since only moving Li ions were observed in the QENS experiment, [23] the fact that no correlations are observed below 4.0 Å confirms that the Li1-type states do not move Li ions, and seem to be trapped.Experimentally, Li ions jump 4.3 Å with a mean residence time τ 0 of 5.5 × 10 −11 s (i.e., 55 ps), and so the AIMD simulation for the smaller timescale does not reproduce this motion; however, the Li3-type charge clouds with correlations at 4.0-5.0Å are distributed in a mesh-like pattern in the glassy structure and are assumed to be constantly cooperating and oscillating, thereby rendering jumping more facile.

Conclusion
Charge topology analysis of the atomic arrangement of glasses that possess a disordered Li-Li correlation was found to reproduce the X-ray and neutron diffraction results, in addition to the electronic states, ultimately revealing three distinct environments for the Li ions.More specifically, the Li3-type charge clouds with correlations at 4.0-5.0Å are distributed in a mesh-like pattern in the glassy structure and are assumed to be constantly cooperating and oscillating, thereby rendering jumping more facile.Therefore, anions with a high polarizability and electronegativity should be combined to construct many Li3-type ions with characteristic correlation lengths.Finally, we consider that the construction of a Li ion transport environment based on molecular vibrations in glasses will be of particular importance, since solid electrolytes should ideally construct a charge situation that facilitates the transport of Li ions.

Theoretical Section
AIMD simulations: The VASP [43,44] was used for the AIMD simulations along with the projector-augmented wave method [45] and the plane-wave basis set.The Perdew-Burke-Ernzerhof (PBE) exchangecorrelation functional [46] for Generalized-Gradient-Approximation (GGA) was also applied in this study.All AIMD simulations were performed using VASP with pseudopotentials (i.e., PAW PBE Li 17Jan2003, PAW PBE P 06Sep2000, and PAW PBE S 06Sep2000) [47,48] and without spin polarization.A kinetic energy cutoff of 517.4 eV was included, and the Brillouin zone was sampled with The initial glass structure for the AIMD simulations was built based on previously reported experimental results. [14]This system contained 208 atoms (i.e., 70 Li, 30 P, and 108 S atoms), which corresponds to the composition of Li 7 P 3 S 11 glass with a 2S deficiency.This 2S deficiency agrees with experimental observations recorded by means of the inductively coupled plasma (ICP) analysis. [14]The volume was 16.1083833A 3 with an experimental density of 1.938 g cm −3 at room temperature.
The crystal structures of γ-Li 3 PS 4, [34] β-Li 3 PS 4, [34] and Li 6 PS 5 Cl [49] were obtained from the crystallographic data listed in the Inorganic Crystal Structure Database. [50]These crystalline phases of Li 3 PS 4 contained 128 atoms (i.e., 48 Li, 16 P, and 64 S atoms).β-Li 3 PS 4 is a high-temperature phase, meaning that it consists of different structures at room temperature, i.e., γ-Li 3 PS 4 and the orthorhombic Li 7 PS 6 .The crystalline phase of Li 6 PS 5 Cl contained 208 atoms (i.e., 96 Li, 16 P, 80 S, and 16 Cl atoms).This structure was constrained to have S 2− and Cl − equally present at the 4a and 4d sites.In addition, periodic boundary conditions in three dimensions (i.e., the x-, y-, and z-directions) were employed for all AIMD simulations.
All AIMD data for the phases were sampled based on DFT using the canonical (NVT) ensemble and the Nosé-Hoover thermostat [51,52] with a time step of 2 fs.After an equilibration AIMD run of several 100 steps, a production run was performed for approximately 2000 steps.Despite the phase, no bond breaking was observed in PS 4  3− during the equilibration and production runs.After completion of the equilibrium calculation the atomic configurations were sampled over 800 steps (i.e., 1.6 ps).
][55] A grid size of 0.2 Å was used for each axis.The regions were divided and analyzed based on the zero-flux surface using the topology of electron density, i.e., the three-dimensional maxima, minima, and saddle points, as follows: The valence of the Li + ion was obtained using the Bader charge estimated from the mass of charges separated by the above method, which was calculated by subtracting the number of core electrons (i.e., 2) and the Bader charge from the total number of electrons (i.e., 3).Owing to the maximum charge position in the divided charge cloud, the distance between this position and the nearby Li ion position was also calculated.VESTA was used for the threedimensional display of the charges. [56]an Hove correlation function analysis: The van Hove correlation function was calculated using AIMD simulations.[35] During this analysis, the Li ions were classified into three types based on the distance between the Li ion position and the charge density maxima determined by Bader charge analysis. The istinct part g αβ (r, t) for α-β correlation, can be calculated as follows: where ρ is the average atomic density, N is the number of atoms, r i (t) is the position of the ith atom at time t, and δ(r) is the Dirac delta function.g(r, t) corresponds to the PDF at t → 0. RMC simulations combined with deep learning potentials (RMC-DL): We determined the structural model of glassy Li 3 PS 4 , which reproduces the experimentally observed S(Q) with different ionic conductivities, using the RMC program [57] with DL potentials.In the conventional RMC program, the energy term calculated by the classical potentials has already been included, but there is no opportunity to use different types of potential; thus, we combined this program with other codes to incorporate the DL potential.The potential for Li 3 PS 4 glass was produced by AIMD simulations of four densities (i.e., 1.6, 1.8, 1.9, and 2.0 g cm −3 ) and at five temperatures (i.e., 100, 300, 1000, 1500, and 2000 K) in 10 000 steps to generate sample data for DL of the structure-energy relationship using the VASP code with PBE potentials under NVT conditions. [43]After the AIMD simulation, the DL potential was calculated from the obtained structures and energies using the aenet program. [58]The input parameters for DL were the Chebyshev descriptors, [59] while two hidden layers with 35 nodes (twists) were used for training.The root mean square error for the obtained potential was <3.0 meV atom −1 .The initial structure prior to the RMC-DL simulations was determined using the MD method with a DL potential. [60]The structure created by the Amorphous Cell code of the BIOVIA Materials Studio 2021 software package was relaxed using DL potentials at 300 K.
Inside the RMC-DL algorithm, the following procedure was performed.Initially, 100 movement steps were generated by the conventional RMC algorithm to satisfy the experimental X-ray and neutron S (Q) values.The energy difference between the structures before and after the RMC simulation was then calculated using the DL potential.The decision on whether these RMC movements were accepted was conducted using the same method as that of the molecular MC approach using the energy differences between the models. [61]Following continuous looping of the above procedure, the obtained structure gradually satisfied the experimental S(Q) values with energetically stable conditions.Therefore, the RMC-DL algorithm was shown to avoid the formation of a thermodynamically unfavorable local structure, which is sometimes formed through conventional RMC simulations.For example, Figure S10, Supporting Information, shows the potential energy trajectory of the glassy structure after 5000 steps.The figure shows that the RMC simulation without DL potentials leads to unstable energies, even when the structural model satisfies the experimental data.On the contrary, in the RMC simulation with DL potentials, although the energy also becomes unstable initially due to the strong demand for satisfying the experimental data, the effect of the energy term becomes stronger and the stable structure gradually reforms as the difference between the structural model and the experimental data becomes smaller during subsequent steps.Thus, a structural model satisfying both the experimental S(Q) values and energies can be generated by the RMC-DL algorithm.

Experimental Section
Synthesis of the glassy sulfide electrolytes: Sulfide electrolyte Li 3 PS 4 was synthesized by a mechanochemical process using a planetary ball mill as described previously. [62]Glassy Li 3 PS 4 and the partially crystallized glassy Li 3 PS 4 were synthesized by milling at 510 rpm for rotation times of 116 and 130 h, respectively.Crystalline βand γ-Li 3 PS 4 phases were synthesized by heat-treatment of glassy Li 3 PS 4 at 250 °C and 480 °C for 2 h, respectively.
Characterization: The ionic conductivity was measured by the AC impedance method in an Ar atmosphere at room temperature with an applied frequency range of 0.1 Hz to 1 MHz using a Solartron 1260 frequency response analyzer.Gold thin-film as an ion-blocking electrode was applied on both sides of the pelletized sample.Synchrotron-based total X-ray scattering measurements with PDF analysis were performed with an incident X-ray energy of 61.4 keV at the BL04B2 beamline in SPring-8, Japan.The data were collected using Ge and CdTe hybrid detectors.The reduced PDF G(r) was obtained using the conventional Fourier transform of the Faber-Ziman structure factor S(Q) [63] extracted from the collected data. [64,65]The time-of-flight neutron total scattering measurements were performed at room temperature using a NOVA spectrometer at BL21 of the MLF in J-PARC, Japan.The sample was contained in a standard cylindrical vanadium can with an outer diameter of 3.0 mm and a wall thickness of 0.1 mm.The data were corrected for the background, sample absorption, cell absorption, multiple scattering, and incoherent scattering using the nvaSq software coded by the NOVA group.To obtain the DF images and electron diffraction patterns, TEM observations were carried out using a JEM-2100F fieldemission-type TEM system.The samples were mounted on an amorphous carbon film supported by a Cu grid, which was then attached to a TEM vacuum holder (Gatan model 648) in a glove box filled with Ar gas.The vacuum degree was ~1.0 × 10 −5 Pa.

Figure 1 .
Figure 1.a) Trajectories of the Li + cations and PS 4 3− anions every 0.2 s between 1.6 ps, as obtained from the AIMD simulations at 300 K. Red, Li; blue, P and PS polyhedral anions; green, S. The coordinates for b) S libration and c) Li vibration of the Li 2 S-P 2 S 5 glass, respectively, showing the transition from the initial position.

Figure 2 .
Figure 2. Environment of the Li ions as evaluated by Bader charge analysis.a) Charge density of the Li ions obtained from Bader analysis of the Li 2 S-P 2 S 5 crystal.b) Mean valence and variance during 1.6 ps for the Li ions.c) Correlation between the mean valence and the variance of Li ions in the glassy, β-, and γcrystal phases.d) Distance obtained by Bader analysis, and e) mean distance between the Li ion position and the charge density maxima over 1.6 ps.f) Correlation between the mean distance and the variance between the Li ion position and the charge density maxima in the glassy, β-, and γ-crystal phases.

Figure 4 .
Figure 4. van Hove correlation function, g(r, t) of the Li ions in the glassy phase.a) the Li1-Li1 correlation, b) the Li2-Li2 correlation, and c) the Li3-Li3 correlation.d) The corresponding time-integrated van Hove correlations.e) Comparison of the Li3-Li3 correlations for the glassy, β-, and γ-crystal phases.

Figure 5 .
Figure 5. Schematic of the charge luck and Li oscillation of each classified Li ion.

Figure 6 .
Figure 6.a) Total structure factors S(Q) for Li 3 PS 4 : black line, alpha crystals; green line, partially crystallized glass; blue line, extracted glassy and interface phases; black dotted line, pure glass.b) Their reduced PDF G(r) values.The colors correspond to those defined for Figure 6a.TEM observations of the partially crystallized Li 3 PS 4 glass.c) DF image of the partially crystallized Li 3 PS 4 glass.d) Selected area electron diffraction pattern corresponding to the DF image.

Figure 7 .
Figure 7. a) RMC-DL model of the partially crystallized glassy phase of Li 3 PS 4 .Red, Li; blue, P and PS polyhedral anions; green, S. b) Total structure factors S(Q) at room temperature for the glassy phase, as derived from the X-ray and neutron diffraction experiments.Circles, experimental data; lines, RMC-DL model.c) Fraction of each Li classified by the Bader method.d) Visualization of the Li3-type Li ions and correlations at 4.0-5.0Å in the constructed glass and the interface structure model.Red, Li (Li3-type); blue, P and PS polyhedral anions.