#
Computational Characterization of β-Li_{3}PS_{4} Solid Electrolyte: From Bulk and Surfaces to Nanocrystals

^{1}

^{2}

^{*}

## Abstract

**:**

_{3}PS

_{4}is emerging as a good solid-state electrolyte candidate due to its stability and ionic conductivity. Despite the number of recent studies on this material, there is still much to understand about its atomic structure, and in particular its surface, a topic that becomes of key relevance for ionic diffusion and chemical stability in grain borders and contact with the other device components. In this study, we performed a density functional study of the structural and electronic properties of β-Li

_{3}PS

_{4}surfaces. Starting from the bulk, we first verified that the thermodynamically stable structure featured slight distortion to the structure. Then, the surfaces were cut along different crystallographic planes and compared with each other. The (100) surface is confirmed as the most stable at T = 298 K, closely followed by (011), (010), and (210). Finally, from the computed surface energies, the Wulff nanocrystals were obtained and it was verified that the growth along the (100) and (011) directions reasonably reproduces the shape of the experimentally observed nanocrystal. With this study, we demonstrate that there are other surfaces besides (100) that are stable and can form interfaces with other components of the battery as well as facilitate the Li-migration according to their porous structures.

## 1. Introduction

_{3}PS

_{4}, stands out due to its good stability, atomic structure, ionic conductivity, and good chemical compatibility. At normal pressure and low temperature, Li

_{3}PS

_{4}is formed as a γ-structure (space group Pmn2

_{1}), with hexagonal close-packed sulfide ion arrays in which the phosphorus ions are distributed over tetrahedral sites and the PS

_{4}tetrahedrons are isolated from each other. With a slow cooling process from 900 °C and crystallizing to 520 °C [3], the γ is converted into a β-structure (space group Pnma), and by increasing the temperature, the phase transition occurs and the α is achieved (which belongs to a supergroup of Pnma) [4].

^{+}ions, the β-structure is the most interesting one with a value of 3.0 × 10

^{−2}S cm

^{−1}at 573 K (or 8.93 × 10

^{−7}S cm

^{−1}by extrapolating to room temperature) [2]. In 2013, Liu et al. [5] synthesized the nanoporous β-Li

_{3}PS

_{4}, which showed an ionic conductivity three orders of magnitude higher than that of the bulk (1.64 × 10

^{−4}S cm

^{−1}at room temperature). The ionic conductivity is favored by a fractional occupation of the sites, which does not occur in the γ structure, and which can explain the best conductivity in the β phase. Recently, Marchini et al. [6] revealed that the great conductivity of the nanoporous β-Li

_{3}PS

_{4}is due to the portion of tetrahydrofuran (THF) that remains after the crystallization.

_{3}PS

_{4}surfaces. However, to the best of our knowledge, a study on the different surfaces of the β-Li

_{3}PS

_{4}, their stabilities, and properties has not been carried out. For this reason, we have attempted detailed first-principle modeling of the β-Li

_{3}PS

_{4}surfaces by computing their structural and electronic properties. According to the calculated surface energies, Wulff nanocrystal morphologies are proposed taking into account the increase or decrease of the stability of each surface.

## 2. Materials and Methods

_{3}PS

_{4}bulk. In particular, the hybrid PBE0 functional [8,9] was adopted and the 6–11G [10], 86–311G* [11], and 85–211dG [12] basis sets were used to describe the Li, S, and P atoms, respectively.

_{BCP}[20]. Put simply, interactions can be subdivided in (i) covalent, if the Laplacian is negative and H(r) and V(r)/G(r) > 2, i.e., if there is an excess of potential energy at the BCP; (ii) transit, if the Laplacian is positive, the bond degree is close to zero and 1 < V(r)/G(r) < 2; (iii) ionic and multipolar (i.e., hydrogen bonds and van der Waals interactions) characterized by positive Laplacian and H(r) and V(r)/G(r) < 1 due to the dominance of kinetic energy at the BCP [21,22].

_{3}PS

_{4}belong to the space group Pnma, with experimental cell parameters [2] a = 12.82 Å, b = 8.22 Å, and c = 6.12 Å. Starting from the fully relaxed bulk structure, surfaces were obtained by cutting along suitable crystallographic planes. According to the experimental powder diffraction pattern, the planes (221) and (210) have a higher intensity and these surfaces were generated in addition to the usual ones: (001), (100), (110), (010), (101), (011), and (111).

_{3}PS

_{4}bulk contains 4 units of Li

_{3}PS

_{4}in the reference cell (32 atoms in total), the number of layers of each surface was chosen to be multiple of 4. Therefore, starting from the minimal 32-atom unit, slabs with an increasing number of layers were considered. For each surface, different terminations along the ±z-direction were analyzed in order to consider different types of low coordination atoms and surface morphologies (see Table S4). The results for each surface, considering the different thicknesses and possible terminations, are discussed in Topic S2 of the Supplementary Information (SI) and shown in Figures S1–S9.

_{surf}), and formation energies (E

_{form}), calculated as ${E}_{surf}=\frac{\left({E}_{slab}^{opt}-n{E}_{bulk}^{opt}\right)}{2\times A}$, and ${E}_{form}=\left({E}_{bulk}^{opt}-{E}_{slab}^{ghost}\right)+\left({E}_{slab}^{no-ghost}-{E}_{slab}^{opt}\right)$, where ${E}_{slab}^{opt}$ and ${E}_{bulk}^{opt}$ are the energy of the optimized surface (slab) and bulk, respectively, $n$ is the number of bulk units in the slab, $A$ is the surface area, ${E}_{slab}^{ghost}$ and ${E}_{slab}^{no-ghost}$ are the energies of the non-optimized surfaces with and without ghost functions. In E

_{form}, the basis set superposition error (BSSE) is incorporated, whose value is related to the atoms that belong to different structures approaching each other, interacting, and the overlap among their basis functions produces a not-genuine stabilization that is inversely proportional to the quality of the adopted basis set. In this work, a posteriori counterpoise (CP) method [23] was applied.

## 3. Results

#### 3.1. Bulk Analysis

_{3}PS

_{4}phase as an electrolyte is probably linked to a residual configurational disorder on the atomic scale. In the Pnma structure, the lithium atoms can occupy three distinct positions, corresponding to Wyckoff 8d, 4b, and 4c symmetry, for a total of 16 sites in which to place the 12 Li of the reference cell. At the macroscopic level, this translates into fractional occupation numbers, which experimentally, at finite temperature, proved to be 1.0 for 8d, 0.7 for 4b, and 0.3 for 4c. Since from the computational point of view it is necessary to maintain the symmetry of the space group, we have adopted the most probable configuration, occupying positions d and b, in line with most of the other theoretical studies [24,25].

_{1}a crystal, a subgroup of Pnma, with lattice parameters a = 12.91 Å, b = 8.14 Å, and c = 6.23 Å, corresponding to a deviation of 0.70%, −0.97 %, and 1.80% from the experimental values of Homma et al. [2]. Its electron energy per cell was 0.09 eV lower than that of the Pnma system and the computed frequencies were then all positive. According to our results, in the Pnma, Li atoms have coordinates similar to those measured by Homma et al. [2] and computed by Yang and coworkers [24], while in the Pn2

_{1}a model they occupy 4a Wickoff positions, similar to those in the distorted B3C1 structure recently calculated by Lim et al. [26], see Table 1.

_{1}a models, see Table 2. The four almost equivalent P-S bonds can be classified as covalent interactions based on the topological values in r = r

_{BCP}: i.e., a high charge density, the negative values of the Laplacian, and a V/G ratio greater than 3. The four Li-S bonds have a so-called transitory nature, sharing features with both the ionic (positive values of the Laplacian in r

_{BCP}) and covalent interactions (cylindrical shape of the charge density around the bond axes and directionality). The small differences found in the topological indicators are due to slight discrepancies in the first coordination spheres of Li atoms at different Wyckoff positions. Based on this analysis, it can be concluded that the Li-S interactions are weak and can be easily broken, leaving the lithium ion free to migrate within the porous material. This obviously plays an important part in Li diffusion.

_{1}a are close to literature data [24], see Table S2. In particular, the β-structure is considered a soft material and this is a property of great interest in the case of SSE devices as the volume of the various components varies during battery operation, and any rigidity would make the structure unstable. A close comparison shows that the Pn2

_{1}a presents Young and Bulk moduli greater than those of the Pnma, highlighting a slightly higher stiffness, but it remains an extremely deformable system.

_{1}a, being a global minimum in the potential energy surface.

_{3}PS

_{4}Pn2

_{1}a model is an insulator with an indirect band gap of 4.75 eV between the Γ and Z points. This value, computed at the PBE0 level, is in fairly good agreement with the one measured by Rangasamy et al. [27] of 5 eV and, as expected, is slightly overestimated when compared with those calculated in a previous theoretical study at the GGA-PBE and HSE06 [24] level. The band structure, reported in Figure 1, is very similar to those proposed by Lim and co-workers for the B3C1 and B4C0 systems [26]. From a qualitative point of view, the presence of wide, well-dispersed bands indicate small effective masses, band velocities, and large electron mobility. The projected density of states (DOS) shows that the low-energy electronic transitions occur between states that belong mainly to the 2p orbitals of the sulfur atoms while Li and P provide an almost negligible contribution to the bands around the Fermi level.

^{−1}, Figure 2, represents the characteristic signal of the β-structure, measured at 422 cm

^{−1}due to the collective vibrations of the (PS

_{4}) units. The wide band between 150 and 300 cm

^{−1}contains the frequencies of all Li-S modes while weak signals above 500 cm

^{−1}are assigned to the P-S bonds.

#### 3.2. β-Li_{3}PS_{4} Surfaces

_{4}) represents an essential structural unit to maintain surface stability, so in modeling the various slabs we tried to preserve the integrity of these units as much as possible. Then, for each crystallographic plane, we designed slabs of increasing thickness, starting from the minimum reference cell, which contains—as in the bulk—four units of Li

_{3}PS

_{4}. We explored planes corresponding to the following Miller indices: (100), (010), (001), (110), (011), (111), (210), (211), according to the distribution density of the diffraction peaks as reported in Ref. [29], and, for each surface, different termination patterns, as LiS

_{3}, PS

_{3}, PS

_{2}, SLi

_{2}, SPLi

_{2}, LiS

_{2}, SPLi, and LiS, were modeled. All the results are collected and described in the Supplementary Information.

_{3}PS

_{4}, are reported in Table 3, while the initial and fully relaxed structures of the four most stable slabs, namely the (100), (010), (011), and (210), are shown in Figure 3.

_{2}-(100) surface, which is the most stable, experiences minimal distortion.

_{surf}, see Table 3, it is possible to derive the following ranking: (100) < (210) < (011) < (211) < (111) < (010) < (001) < (110). The correction due to the BSSE error produces a slightly different order, (100) < (010) = (011) < (210) < (111) < (210) < (110) = (001), but the extremes remain the same: the surface (100) is the easiest to obtain, with a low E

_{surf}and E

_{form}close to zero, and the (110) surface is the most difficult, with high positive values of both E

_{surf}and E

_{form}. Then, the surface Gibbs free energy (G, expressed in kJ/(mol·m²)) is calculated for the four structures with the lowest surface energy, namely (100), (010), (011), and (210), and (100), G = 1.96 × 10

^{−19}, is confirmed as the most stable surface, followed by (210) (G = 2.11 × 10

^{−19}), (011) (G = 3.13 × 10

^{−19}), and (010) (G = 3.95 × 10

^{−19}).

_{3}PS

_{4}nanocrystals are not only plausible but desirable due to the new properties they may have.

## 4. Conclusions

_{3}PS

_{4}bulk and surface stability was conducted using the CRYSTAL program, and by applying all-electron basis-sets and the PBE0 functional. This study mainly aims to investigate the stability of the 2D thin film of β-Li

_{3}PS

_{4}for their use in lithium batteries as a solid electrolyte.

_{1}a subgroup, in which the P-S lattice remains unchanged while the 12 Li atoms occupy three different Wyckoff positions: 4a′, 4a″, and 4a′′′. Then, a set of slabs, whose thickness varies between 9 to 23 Å, were modeled by cutting the bulk along different crystallographic directions. Based on the calculated surface formation energies and Gibbs free energies, the (100) surface resulted as the most stable, followed by (210), (011), and (010). In addition, the (100) and (210) films showed interesting features such as good mechanical and thermodynamic stability, a high concentration and good mobility of Li ions (i.e., the Li atoms are loosely bonded and the lattice is extremely porous), and a potentially small lattice mismatch with materials such as Li

_{2}S, widely used as a passivator in lithium technology. The (010) structure presents original characteristics, such as peculiar surface states and a net dipole moment along the non-periodic z-axis, which could eventually increase the diffusion rate of the lithium ions. Finally, Wulff’s theory was applied to derive the shapes of possible nanocrystals. As proof, it has been verified that the extra-stabilization of both the (100) and (011) surfaces, done by hand, leads to nanoplates very similar to those synthesized experimentally.

## Supplementary Materials

_{3}PS

_{4}; Table S2: Mechanical properties of β-Li

_{3}PS

_{4}; Table S3: Possible termination along +z of each analyzed surface. The “X” represents the termination found in the surface and the ‘-’ the termination not found; Table S4: Hirshfield charges of β-Li

_{3}PS

_{4}bulk and eight-unit surfaces for the surface (indicated by *) and internal atoms; Figure S1: (a) SLi

_{2}/SP and (b) LiS

_{2}/LiS

_{2}; Figure S2: (a) SLi

_{2}/SLi, (b) PS

_{3}/SP, (c) LiS

_{2}/LiS

_{2}, and (d) LiS/PS; Figure S3: (a) LiS

_{2}/Li and (b) PS

_{3}/SLi

_{2}; Figure S4: (a) PS

_{3}/LiS, (b) PS

_{3}/SPLi, (c) LiS

_{2}/PS

_{2}, and (d) LiS/LiS; Figure S5: (a) PS

_{3}/Li and (b) LiS

_{2}/Li; Figure S6: (a) SLi

_{2}/Li, (b) LiS/LiS, and (c) PS

_{2}/SPLi; Figure S7: (a) LiS

_{3}/LiS, (b) PS

_{2}/LiS

_{2}, (c) SLi

_{2}/LiS

_{2}, and (d) Li/PS

_{3}; Figure S8: (a) SPLi/Li, (b) LiS

_{2}/LiS, and (c) LiS

_{2}/SLi

_{2}; Figure S9: (a) SLi

_{2}/LiS, (b) LiS

_{2}/SPLi, and (c) PS

_{2}/LiS; Figure S10: Three-dimensional maps of the electronic charge density superimposed to the electrostatic potential of β-Li

_{3}PS

_{4}surfaces along z-direction: (a) (100), (b) (210), and (c) (011). The spheres in purple, orange, yellow, and black are related to the lithium, phosphor, and sulfur atoms. The scale maps range from negative (−) in blue to positive (+) in red. References [2,24,25] were citied in the Supplementary Materials.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ahniyaz, A.; de Meatza, I.; Kvasha, A.; Garcia-Calvo, O.; Ahmed, I.; Sgroi, M.F.; Giuliano, M.; Dotoli, M.; Dumitrescu, M.A.; Jahn, M.; et al. Progress in Solid-State High Voltage Lithium-Ion Battery Electrolytes. Adv. Appl. Energy
**2021**, 4, 100070. [Google Scholar] [CrossRef] - Homma, K.; Yonemura, M.; Kobayashi, T.; Nagao, M.; Hirayama, M.; Kanno, R. Crystal Structure and Phase Transitions of the Lithium Ionic Conductor Li
_{3}PS_{4}. Solid State Ion.**2011**, 182, 53–58. [Google Scholar] [CrossRef] - Mercier, R.; Malugani, J.-P.; Fahys, B.; Robert, G.; Douglade, J. IUCr Structure Du Tetrathiophosphate de Lithium. Acta Crystallogr. Sect. B-Struct. Sci.
**1982**, 38, 1887–1890. [Google Scholar] [CrossRef] - Kim, J.-S.; Jung, W.D.; Choi, S.; Son, J.-W.; Kim, B.-K.; Lee, J.-H.; Kim, H. Thermally Induced S-Sublattice Transition of Li
_{3}PS_{4}for Fast Lithium-Ion Conduction. J. Phys. Chem. Lett.**2018**, 9, 5592–5597. [Google Scholar] [CrossRef] [PubMed] - Liu, Z.; Fu, W.; Payzant, E.A.; Yu, X.; Wu, Z.; Dudney, N.J.; Kiggans, J.; Hong, K.; Rondinone, A.J.; Liang, C. Anomalous High Ionic Conductivity of Nanoporous β-Li
_{3}PS_{4}. J. Am. Chem. Soc.**2013**, 135, 975–978. [Google Scholar] [CrossRef] - Marchini, F.; Porcheron, B.; Rousse, G.; Blanquer, L.A.; Droguet, L.; Foix, D.; Koç, T.; Deschamps, M.; Tarascon, J.M. The Hidden Side of Nanoporous β-Li
_{3}PS_{4}Solid Electrolyte. Adv. Energy Mater.**2021**, 11, 2101111. [Google Scholar] [CrossRef] - Dovesi, R.; Erba, A.; Orlando, R.; Zicovich-Wilson, C.M.; Civalleri, B.; Maschio, L.; Rérat, M.; Casassa, S.; Baima, J.; Salustro, S.; et al. Quantum-Mechanical Condensed Matter Simulations with CRYSTAL. Wiley Interdiscip. Rev. Comput. Mol. Sci.
**2018**, 8, e1360. [Google Scholar] [CrossRef] - Perdew, J.P.; Wang, Y. Accurate and Simple Analytic Representation of the Electron-Gas Correlation Energy. Phys. Rev. B
**1992**, 45, 13244. [Google Scholar] [CrossRef] [PubMed] - Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0 Model. J. Chem. Phys.
**1999**, 110, 6158. [Google Scholar] [CrossRef] - Dovesi, R.; Ermondi, C.; Ferrero, E.; Pisani, C.; Roetti, C. Hartree-Fock Study of Lithium Hydride with the Use of a Polarizable Basis Set. Phys. Rev. B
**1984**, 29, 3591. [Google Scholar] [CrossRef] - Lichanot, A.; Aprà, E.; Dovesi, R. Quantum Mechnical Hartree-Fock Study of the Elastic Properties of Li2S and Na2S. Phys. Status Solidi
**1993**, 177, 157–163. [Google Scholar] [CrossRef] - Zicovich-Wilson, C.M.; Bert, A.; Roetti, C.; Dovesi, R.; Saunders, V.R. Characterization of the Electronic Structure of Crystalline Compounds through Their Localized Wannier Functions. J. Chem. Phys.
**2001**, 116, 1120. [Google Scholar] [CrossRef] - Dovesi, R.; Saunders, V.; Roetti, C.; Orlando, R.; Zicovich-Wilson, C.M.; Pascale, F.; Civalleri, B.; Doll, K.; Harrison, N.; Bush, I.; et al. CRYSTAL17 User’s Manual. 2018. Available online: https://www.crystal.unito.it/manuals/crystal17.pdf (accessed on 14 July 2022).
- Pascale, F.; Zicovich-Wilson, C.M.; Gejo, F.L.; Civalleri, B.; Orlando, R.; Dovesi, R. The Calculation of the Vibrational Frequencies of Crystalline Compounds and Its Implementation in the CRYSTAL Code. J. Comput. Chem.
**2004**, 25, 888–897. [Google Scholar] [CrossRef] - Zicovich-Wilson, C.M.; Pascale, F.; Roetti, C.; Saunders, V.R.; Orlando, R.; Dovesi, R. Calculation of the Vibration Frequencies of α-Quartz: The Effect of Hamiltonian and Basis Set. J. Comput. Chem.
**2004**, 25, 1873–1881. [Google Scholar] [CrossRef] - Bader, R.F.W. Atoms in Molecules: A Quantum Theory; Oxford University Press: Oxford, UK, 1990; p. 438. [Google Scholar]
- Casassa, S.; Erba, A.; Baima, J.; Orlando, R. Electron Density Analysis of Large (Molecular and Periodic) Systems: A Parallel Implementation. J. Comput. Chem.
**2015**, 36, 1940–1946. [Google Scholar] [CrossRef] [PubMed] - Gatti, C.; Casassa, S. TOPOND14 User’s Manual. 2017, p. 53. Available online: https://www.crystal.unito.it/topond/topond.pdf (accessed on 14 July 2022).
- Gatti, C.; Saunders, V.R.; Roetti, C. Crystal Field Effects on the Topological Properties of the Electron Density in Molecular Crystals: The Case of Urea. J. Chem. Phys.
**1994**, 101, 10686–10696. [Google Scholar] [CrossRef] - Gatti, C. Chemical Bonding in Crystals: New Directions. Z. Krist.
**2005**, 220, 399–457. [Google Scholar] [CrossRef] - Gatti, C. Challenging Chemical Concepts through Charge Density of Molecules and Crystals. Phys. Scr.
**2013**, 87, 48102. [Google Scholar] [CrossRef] - Clark, A.E.; Sonnenberg, J.L.; Hay, P.J.; Martin, R.L. Density and Wave Function Analysis of Actinide Complexes: What Can Fuzzy Atom, Atoms-in-Molecules, Mulliken, Löwdin, and Natural Population Analysis Tell Us? J. Chem. Phys.
**2004**, 121, 2563–2570. [Google Scholar] [CrossRef] - van Duijneveldt, F.B.; van Duijneveldt-van de Rijdt, J.G.C.M.; van Lenthe, J.H. State of the Art in Counterpoise Theory. Chem. Rev.
**1994**, 94, 1873–1885. [Google Scholar] [CrossRef] - Yang, Y.; Wu, Q.; Cui, Y.; Chen, Y.; Shi, S.; Wang, R.Z.; Yan, H. Elastic Properties, Defect Thermodynamics, Electrochemical Window, Phase Stability, and Li
^{+}Mobility of Li_{3}PS_{4}: Insights from First-Principles Calculations. ACS Appl. Mater. Interfaces**2016**, 8, 25229–25242. [Google Scholar] [CrossRef] [PubMed] - Lepley, N.D.; Holzwarth, N.A.W.; Du, Y.A. Structures, Li+ Mobilities, and Interfacial Properties of Solid Electrolytes Li
_{3}PS_{4}and Li3PO4 from First Principles. Phys. Rev. B**2013**, 88, 104103. [Google Scholar] [CrossRef] - Lim, M.S.; Jhi, S.H. First-Principles Study of Lithium-Ion Diffusion in β-Li
_{3}PS_{4}for Solid-State Electrolytes. Curr. Appl. Phys.**2018**, 18, 541–545. [Google Scholar] [CrossRef] - Rangasamy, E.; Li, J.; Sahu, G.; Dudney, N.; Liang, C. Pushing the Theoretical Limit of Li-CFx Batteries: A Tale of Bifunctional Electrolyte. J. Am. Chem. Soc.
**2014**, 136, 6874–6877. [Google Scholar] [CrossRef] - Ates, T.; Neumann, A.; Danner, T.; Latz, A.; Zarrabeitia, M.; Stepien, D.; Varzi, A.; Passerini, S.; Ates, T.; Neumann, A.; et al. Elucidating the Role of Microstructure in Thiophosphate Electrolytes—A Combined Experimental and Theoretical Study of β-Li
_{3}PS_{4}. Adv. Sci.**2022**, 9, 2105234. [Google Scholar] [CrossRef] - Wang, H.; Hood, Z.D.; Xia, Y.; Liang, C. Fabrication of Ultrathin Solid Electrolyte Membranes of β-Li
_{3}PS_{4}Nanoflakes by Evaporation-Induced Self-Assembly for All-Solid-State Batteries. J. Mater. Chem. A**2016**, 4, 8091–8096. [Google Scholar] [CrossRef] - Bultinck, P.; Van Alsenoy, C.; Ayers, P.W.; Carbó-Dorca, R. Critical Analysis and Extension of the Hirshfeld Atoms in Molecules. J. Chem. Phys.
**2007**, 126, 144111. [Google Scholar] [CrossRef] - Hood, Z.D.; Wang, H.; Pandian, A.S.; Peng, R.; Gilroy, K.D.; Chi, M.; Liang, C.; Xia, Y. Fabrication of Sub-Micrometer-Thick Solid Electrolyte Membranes of β-Li
_{3}PS_{4}via Tiled Assembly of Nanoscale, Plate-Like Building Blocks. Adv. Energy Mater.**2018**, 8, 1800014. [Google Scholar] [CrossRef]

**Figure 2.**Raman spectra of β-Li

_{3}PS

_{4}(

**a**) Pn2

_{1}a crystal computed at the PBE0 level and (

**b**) experimental obtained for different samples by Liu and co-workers [5].

**Figure 3.**Stable surfaces cut from bulk, before and after geometry optimization (

**a**) (100), (

**b**) (010), (

**c**) (011), and (

**d**) (210). The spheres in purple, orange, yellow, and black are related to the lithium, phosphor, and sulfur atoms, respectively.

**Figure 4.**Band structure and density of states of (

**a**) (100), (

**b**) (010), (

**c**) (011), and (

**d**) (210). The curves in purple, orange, yellow, and black are related to the lithium, phosphor, sulfur, and total atoms contributions, respectively.

**Figure 5.**Electrostatic potential maps of the β-Li

_{3}PS

_{4}surfaces (

**a**) (100) and (

**b**) (010). Continuous and dashed lines for positive and negative values of the potential.

**Figure 6.**Different morphologies of β-Li

_{3}PS

_{4}nanocrystals obtained by varying the relative stability of the various surfaces. Panel (

**a**): the ideal nanocrystal based on our results. The blue arrows indicate the stabilization of the (100), (011), and (210) surfaces while the curly brackets indicate the simultaneous stabilization of the surfaces (100)/(011), (210)/(100), and (100)/(011). The resulting nanostructures are in the (

**b**–

**d**) panels, respectively. In the (

**d**) panel, the SEM of β-Li

_{3}PS

_{4}obtained by Hood and co-authors is also reported [31].

**Table 1.**Lithium atomic coordinates on β-Li

_{3}PS

_{4}for the Pnma and Pn2

_{1}a structures, compared with literature experimental and theoretical results.

Li-Sites | Pnma | Pn2_{1}a | Experimental [2] | Theoretical [24] | Theoretical B3C1 [26] |
---|---|---|---|---|---|

8d | (0.332, 0.035, 0.388) | - | (0.356, 0.013, 0.439) | (0.333, 0.033, 0.394) | - |

4b | (0.000, 0.000, 0.500) | - | (0.000, 0.000, 0.500) | (0.000, 0.000, 0.500) | - |

4a′ | - | (0.329, 0.043, 0.410) | - | - | (0.330, 0.041, 0.420) |

4a″ | - | (0.665, 0.972, 0.624) | - | - | (0.663, 0.973, 0.627) |

4a′′′ | - | (0.006, 0.045, 0.563) | - | - | (0.007, 0.053, 0.568) |

**Table 2.**Value of several local quantities at the bond critical points in Pn2

_{1}a β-Li

_{3}PS

_{4}structure: the bond distance (Å), the electron charge density ($\rho $(r)) (Å

^{−3}), the Laplacian of the density (${\nabla}^{2}\rho $) (Å

^{−5}), the ratio between the potential energy density and the kinetic density (|V|/G), the bond degree (H/ρ) (a.u.) (i.e., the ratio between the total energy density and electron density), and the ellipticity ε. The indexes are related to the atom label.

d | $\mathit{\rho}\left(\mathbf{r}\right)$ | ${\nabla}^{2}\mathit{\rho}$ | |V|/G | $\mathbf{H}/\mathit{\rho}$ | $\mathit{\epsilon}$ | |
---|---|---|---|---|---|---|

Li_{1}-S_{30} | 2.502 | 0.0162 | 0.0847 | 0.8022 | 0.2156 | 0.0448 |

Li_{2}-S_{17} | 2.456 | 0.0180 | 0.0933 | 0.8177 | 0.1990 | 0.0169 |

Li_{2}-S_{28} | 2.448 | 0.0180 | 0.0911 | 0.8174 | 0.1945 | 0.0402 |

Li_{2}-S_{18} | 2.459 | 0.0171 | 0.0875 | 0.8075 | 0.2060 | 0.0403 |

Li_{2}-S_{29} | 2.483 | 0.0165 | 0.0862 | 0.8034 | 0.2140 | 0.0369 |

Li_{9}-S_{17} | 2.445 | 0.0183 | 0.0925 | 0.8218 | 0.1911 | 0.0202 |

Li_{9}-S_{21} | 2.526 | 0.0152 | 0.0761 | 0.8019 | 0.2066 | 0.0425 |

Li_{9}-S_{32} | 2.612 | 0.0115 | 0.0596 | 0.7508 | 0.2570 | 0.0239 |

Li_{9}-S_{28} | 2.618 | 0.0121 | 0.0615 | 0.7677 | 0.2382 | 0.1412 |

Li_{9}-S_{25} | 2.713 | 0.0099 | 0.0494 | 0.7549 | 0.2440 | 0.3477 |

P_{16}-S_{2} | 2.082 | 0.1369 | −0.1844 | 3.297 | −0.5960 | 0.0225 |

P_{15}-S_{22} | 2.065 | 0.1411 | −0.1942 | 3.286 | −0.6110 | 0.0136 |

P_{13}-S_{24} | 2.054 | 0.1433 | −0.2003 | 3.266 | −0.6210 | 0.0239 |

P_{13}-S_{29} | 2.050 | 0.1439 | −0.1976 | 3.2225 | −0.6240 | 0.0327 |

**Table 3.**Surfaces’ termination (up/down), surface energy (E

_{surf}, in J/m²), formation energy (E

_{form}, in kJ/mol), and band gap (E

_{gap}, in eV) for the nine surfaces of β-Li

_{3}PS

_{4}with eight units each.

Termination | E_{surf} | E_{form} | E_{gap} | |
---|---|---|---|---|

(001) | LiS_{2}/LiS_{2} | 2.15 | 31.94 | 1.82 |

(100) | LiS_{2}/LiS_{2} | 0.91 | 2.19 | 4.66 |

(010) | LiS_{2}/Li | 1.83 | 5.16 | 2.70 |

(101) | PS_{3}/SPLi | 8.20 | 54.10 | 0.46 |

(011) | PS_{3}/Li | 1.43 | 5.24 | 4.24 |

(110) | SLi_{2}/Li | 2.27 | 32.22 | 1.99 |

(111) | LiS_{3}/Li | 1.60 | 10.91 | 2.21 |

(210) | SPLi/Li | 0.99 | 5.88 | 4.20 |

(211) | LiS_{2}/SPLi | 1.57 | 11.84 | 2.14 |

**Table 4.**Hirshfield charges of β-Li

_{3}PS

_{4}Pn2

_{1}a bulk and eight-unit surfaces for the surface (indicated by *) and internal atoms.

Li_{1} | Li*_{1} | Li_{2} | Li*_{2} | P | P* | S_{1} | S*_{1} | S_{2} | S*_{2} | S_{3} | S*_{3} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

bulk | 1.007 | 1.007 | 1.011 | 1.011 | 1.573 | 1.573 | −1.147 | −1.147 | −1.272 | −1.272 | −1.034 | −1.034 |

(001) | 1.006 | 1.004 | 1.008 | 0.993 | 1.580 | 1.127 | −1.150 | −1.023 | −1.249 | −1.206 | −1.059 | −1.176 |

(100) | 1.007 | 1.003 | 1.010 | 1.008 | 1.575 | 1.498 | −1.149 | −1.130 | −1.268 | −1.297 | −1.033 | −1.020 |

(010) | 1.006 | 0.979 | 1.008 | 1.011 | 1.611 | 1.500 | −1.174 | −0.945 | −1.239 | −1.253 | −1.033 | −1.035 |

(011) | 1.009 | 1.001 | 1.013 | 1.012 | 1.566 | 1.588 | −1.226 | −0.974 | −1.261 | −1.022 | −1.183 | −0.836 |

(210) | 1.009 | 0.972 | 1.011 | 1.002 | 1.565 | 1.549 | −1.177 | −1.006 | −1.246 | −1.180 | −1.107 | −1.001 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Marana, N.L.; Sgroi, M.F.; Maschio, L.; Ferrari, A.M.; D’Amore, M.; Casassa, S.
Computational Characterization of β-Li_{3}PS_{4} Solid Electrolyte: From Bulk and Surfaces to Nanocrystals. *Nanomaterials* **2022**, *12*, 2795.
https://doi.org/10.3390/nano12162795

**AMA Style**

Marana NL, Sgroi MF, Maschio L, Ferrari AM, D’Amore M, Casassa S.
Computational Characterization of β-Li_{3}PS_{4} Solid Electrolyte: From Bulk and Surfaces to Nanocrystals. *Nanomaterials*. 2022; 12(16):2795.
https://doi.org/10.3390/nano12162795

**Chicago/Turabian Style**

Marana, Naiara Leticia, Mauro Francesco Sgroi, Lorenzo Maschio, Anna Maria Ferrari, Maddalena D’Amore, and Silvia Casassa.
2022. "Computational Characterization of β-Li_{3}PS_{4} Solid Electrolyte: From Bulk and Surfaces to Nanocrystals" *Nanomaterials* 12, no. 16: 2795.
https://doi.org/10.3390/nano12162795