Computational Study of Li Ion Electrolytes Composed of Li 3 AsS 4 Alloyed with Li 4 GeS 4

First principles computational techniques are used to study properties of promising Li ion electrolytes, recently developed at based on alloys having the composition Li 3 + x As 1 − x Ge x S 4 . The crystal structure of pure Li 3 AsS 4 is found to be characterized by the Pmn 2 1 space group. Based on modiﬁcations of this structure, reasonable models of the x = 1 / 4 and x = 1 / 3 alloys are found to be in good agreement with the experimental X-ray diffraction patterns and to be consistent with the measured trends in Li ion conduction. As a consequence of their Pmn 2 1 -based structures, interstitial and interstitialcy mechanisms are found to be important for the Li ion conduction processes in these systems. The development of stable solid electrolytes with high ionic con- ductivity provides a challenge for basic materials research 1 which has been identiﬁed as a key to improved battery technologies. 2 Recent experimental studies by Sahu et al. 3 explored new electrolyte compositions for Li ion electrolytes and found promising results based on alloys of Li 3 AsS 4 and Li 4 GeS 4 . Signiﬁcantly higher ionic conductivity was found for the alloys Li 3 + x As 1 − x Ge x S 4 with composi- tions

The development of stable solid electrolytes with high ionic conductivity provides a challenge for basic materials research 1 which has been identified as a key to improved battery technologies. 2 Recent experimental studies by Sahu et al. 3 explored new electrolyte compositions for Li ion electrolytes and found promising results based on alloys of Li 3 AsS 4 and Li 4 GeS 4 . Significantly higher ionic conductivity was found for the alloys Li 3+x As 1−x Ge x S 4 with compositions 0 < x < 0.5, compared with the pure materials, especially for x = 1/3. In order to understand the reported experimental results in terms of the stability and ion conduction mechanisms in this system, we have computationally examined this alloy system using first principles methods.
The notion of improving ionic conductivity by preparing solid solutions of two compatible ionic crystalline materials has been discussed several times in the literature. For example, in a 1977 publication, Hu et al. 4 studied a series of compounds with compositions Li 3+x P 1−x Si x O 4 , finding the highest conductivities for x ≈ 0.5. This system has been further examined by several other researchers, including a recent combined experimental and simulation study by Deng et al. 5 who analyzed the detailed structures of the solid solutions in addition to the ionic conductivity from impedance and nuclear magnetic resonance measurements. For this system, in the composition range 0 ≤ x 0.5, the compounds are derived from the structure of γ-Li 3 PO 4 , having the space group Pnma (No. 62 as listed in the International Table of Crystallography 6 ), with Si substituting for P. The simulation studies of this work suggest that the dominant Li ion migration mechanism for this system involves the correlated motions of interstitial and lattice Li ions in "interstitialcy" or "kick-out" processes. Another related alloy system was recently investigated by Hori et al. 7 who determined the phase diagram of Li 3+x P 1−x Ge x S 4 . These authors find that for very small values of x (0 ≤ x ≤ 0.02), the system takes the Pmn2 1 (No. 31) structure of γ-Li 3 PS 4 , but for 0.02 ≤ x < 0.2 the system takes the Pnma (No. 62) structure of β-Li 3 PS 4 . The alloy system having the β-Li 3 PS 4 structure was found to have a room temperature ion conductivity of 10 −3 S/cm for x = 0.2. 8 For larger values of x the system includes the so-called superionic conducting phase 9 for x = 1/3 which is usually written as Li 10 GeP 2 S 12 (LGPS). The LGPS structure is found for values of 0.2 < x < 0.6, while for x > 0.6 the system takes the Pnma structure of Li 4 GeS 4 . From the temperature dependence of the phase diagram, the authors argue that these structures are stabilized by entropy which is effected by the distribution of differently sized building blocks of PS 4 and GeS 4 ions. The LGPS system has been shown 9,10 to have a room temperature Li ion conductivity of 10 −2 S/cm. The importance of the lattice structure in determining the ionic conductivity of these electrolytes was recently shown by Liu et al. 11 who developed a novel preparation method which stabilizes the β-Li 3 PS 4 structure at room temperature and increases the ionic conductivity by a factor of 100 to 10 −4 S/cm. Other preparation methods have found 8,12 that pure β-Li 3 PS 4 is not stable at room temperature, but its high temperature Li ion conductivity extrapolated to room temperature is approximately 10 −6 S/cm. For the Li 3+x As 1−x Ge x S 4 alloy system studied by Sahu et al., 3 which is the subject the present computational study, the X-ray experimental results suggest that the structure remains that of the Li 3 AsS 4 lattice throughout the composition range 0 ≤ x ≤ 0.5. While a more detailed analysis of the X-ray patterns may reveal some structural variations, we based our simulations on the assumption that Ge substitutes for As in the Li 3 AsS 4 structure. The first task of the simulations is to determine the Li 3 AsS 4 structure itself. Secondly, the likely Ge substitutions and their corresponding placements of the extra Li sites must be determined. Once plausible structures for the alloy are determined, the ion migration mechanisms can be investigated for each structure.

Computational Methods
The computational methods used in this work are based on density functional theory (DFT), 13,14 using the projected augmented wave (PAW) 15 formalism. The PAW basis and projector functions were generated by the ATOMPAW 16 code and used in QUANTUM ESPRESSO 17 package. Visualizations were constructed using the XCrySDEN, 18,19 VESTA 20 software packages.
The exchange correlation function is approximated using the local-density approximation (LDA). 21 The choice of LDA functional was made based on previous investigations [22][23][24][25][26][27][28][29][30] of similar materials which showed that provided that the lattice constants are scaled by a correction factor or 1.02, the simulations are in good agreement with experiment, especially lattice vibrational frequencies and heats of formation.
In this work, the heat of formation H f reported here is determined at 0 degrees K as estimated from the total electronic energy of the material per formula unit minus the energies of the elemental constituents in their standard states as defined by the CRC Handbook. 31 For example, the heat of formation for the alloy can be approximated from the zero temperature total energy results according to where E std corresponds to the total energies of Li in the bcc structure, As in the R3m structure, Ge in the diamond structure, and S in the Fddd (α − S 8 ) structure, respectively. Also of interest is the decomposition energy ( H D ) for an alloy which can also be estimated from the zero temperature total energy results according to: [Li 4 GeS 4 ]. [2] With this sign convention, a negative value for H D implies that the alloy is stable with respect to decomposition into the pure materials. These energy estimates neglect the effects of finite temperature and pressure which van de Walle and Ceder 32 estimate to contribute 10 −2 eV/atom at room temperature for most materials.
For analyzing the heats of formation and other perfect crystal properties, calculations were performed with plane wave expansions of the wave function including |k + G| 2 ≤ 64 bohr −2 and with a Brillouin-zone sampling grid density of at least 10 −3 bohr −3 /k point. The partial densities of states were calculated as described in previous work, 26,29 using weighting factors based on the charge within the augmentation spheres of each atom with radii r Li c = 1.6, r As c = 2.3, r Ge c = 2.1, and r S c = 1.7 in bohr units. The reported partial densities of states curves N a (E) were averaged over the atomic sites of each type a.
Li vacancies and interstitials were modeled in supercell containing 16, 16, and 24 formula units for simulating pure Li 3 AsS 4 , the alloy Li 13/4 As 3/4 Ge 1/4 S 4 , and the alloy Li 10/3 As 2/3 Ge 1/3 S 4 , respectively. For the case of Li 10/3 As 2/3 Ge 1/3 S 4 , the plane wave cut off was reduced to 49 bohr −2 . The estimated error for the relative energies computed with the reduced plane wave cutoff is less than 0.02 eV. These supercells were used to estimate the minimum energy path for Li ion migration using the "nudged elastic band" (NEB) method [33][34][35] as programmed in the QUANTUM ESPRESSO package, using 5 images between each metastable configuration. In modeling charged defects (Li ion vacancies or interstitials), the system was assumed to remain electrically insulating and a uniform background charge was added in order to evaluate the electrostatic interactions. For each minimum energy path, the migration energy, E m was determined as the energy difference between the lowest and highest energy of the path. The "formation energies" E f for producing neutral defects in the form of vacancy-interstitial pairs were calculated for the same supercells. These energies are related to the experimentally measured ionic conductivity through an Arrhenius relationship of the form where A denotes a constant, T denotes the temperature in Kelvin, k denotes the Boltzmann constant, and E A denotes the activation energy for ion migration. The activation energy is related to the migration and formation energies according to 36 In Eq. 4 the upper limit of E A applies to the "intrinsic" case when ion migration must be preceded by the creation of a pair of vacancy and interstitial ions by thermal activation. The lower limit applies to the "extrinsic" case when a population of vacancy or interstitial ions is available so that the conduction depends only the thermal activation due to the migration energy barriers characterized by E m . In principle, the energy of charged defects should be corrected for finite size errors by using one of several methods reported in the literature. 37,38 This was not done in the present work, although we expect that the finite size errors for energy difference such as E m benefit from error cancellation.

Results
Structures of the pure materials.-To the best of our knowledge, the crystal structure of Li 3 AsS 4 has not been analyzed, although its X-ray diffraction pattern is reported by Sahu et al. 3 It is reasonable to assume that Li 3 AsS 4 has structural properties similar to those of Li 3 PS 4 , which has been well studied. 12,26,[39][40][41][42] As mentioned in the introduction, Li 3 PS 4 is known to have the low temperature structure, γ-Li 3 PS 4 , characterized by the space group Pmn2 1 (No. 31). At temperatures between 200 and 300 degrees C, a transition to the β-Li 3 PS 4 structure has been reported, 12,41 which is characterized by the space group Pnma (No. 62) with fractional occupancy of some of the Li sites. The β-Li 3 PS 4 structure can be stabilized at room temperature and has been shown to have higher Li ion conductivity than the γ phase. 40 In previous work, we simulated the ideal crystalline structures of four models of Li 3 PS 4 based on the ordered structures of Pmn2 1 γ-Li 3 PS 4 , Pnma γ-Li 3 PO 4 , and two idealizations of disordered Pnma β-Li 3 PS 4 . 26 The idealizations of disordered Pnma β-Li 3 PS 4 were labeled β-Li 3 PS 4 -b and β-Li 3 PS 4 -c representing models with 100% occupancy of the Li sites with Wyckoff labels b and c, respectively. More recently, Ganesh et al. 39 have shown that the disordered β-Li 3 PS 4 structure can be explained by using molecular dynamics (MD) simulations which show that the 70% occupancy of the b site and 30% occupancy of the c site correlates with the site analysis of the time averaged Li MD trajectories. Table I lists the heats of formation ( H f ) calculated for four structural models structures of Li 3 AsS 4 and Li 3 PS 4 . The heat of of formation results given in Table I, indicate that the ground state structure of Li 3 AsS 4 corresponds to the Pmn2 1 ground state structure of γ-Li 3 PS 4 . The calculated lattice constants are a = 7.65 Å, b = 6.54 Å, c = 6.14 Å, which agrees well with experiment when scaled by the 1.02 LDA correction factor. A structural diagram of Li 3 AsS 4 in this structure is shown in Fig. 1. The calculated fractional coordinates are listed in Table AI in the Appendix.
In order to compare the simulation results with experiment, we generated the X-ray powder diffraction pattern with the help of the Mercury 43 software package and compared it with the experimental pattern, obtained by digitizing the published results, 3 as shown in Fig. 2. The simulated powder pattern is seen to be in good agreement with the experimental results. Also included in Fig. 2 is the  simulated powder pattern for the so-called β-Li 3 PS 4 -c model structure of Li 3 AsS 4 , which has less favorable correspondence to the experimental pattern, consistent with its calculated total energy being 0.14 eV above that of the ground state configuration.
From the simulation results and from the reported X-ray diffraction experiments on Li 3 PS 4 and its alloys, 44 it is reasonable to assume that the Pmn2 1 structure forms the basis of the structure throughout the composition range. That is, unlike Li 3 PS 4 , there is no evidence of multiple structural phases.
In anticipation of the analysis of Li ion migration in this material, it is useful to study the relative energies of vacancy and interstitial defects. The Pmn2 1 structure of Li 3 AsS 4 has two geometrically distinct host lattice sites with Wyckoff labels a and b as indicated in Fig. 1. We found several metastable interstitial sites in the lattice; the lowest energy sites are labeled I (having a multiplicity of 2 in the conventional unit cell) and I I (having a multiplicity of 4). These sites and their equivalents are shown in Fig. 1. The relative energies of vacancies in the 16 formula unit supercell are given in Table II. Interestingly, the relative energies and fractional coordinates of the vacancy and interstitial sites are calculated to be very similar for Li 3 AsS 4 and its γ-Li 3 PS 4 analog.
The crystal structure of Li 4 GeS 4 , which has been studied by several groups, [45][46][47] has the Pnma (No. 62) space group. The calculated lattice Table II. List of relative energies E (eV) for vacancy (Vac) and interstitial (Inter) Li sites in Li 3 AsS 4 compared with the corresponding defects in γ-Li 3 PS 4 calculated for 2×2×2 supercell models. The zero energies are taken to be the b site for the vacancy defect and the I site for the interstitial defect. The "Position" entries indicate the fractional coordinates referenced to the conventional cell of the perfect lattice. Structures of the alloys.-In order to perform detailed first principles simulations of the alloy structures, it is necessary to find plausible ordered structures within computationally manageable supercells as described below. We also briefly investigated the question of whether the material produced in the experiment is likely to be an ordered structure such as found in our simulations, or more likely to be a disordered alloy with Ge replacing As randomly. In the X-ray power diffraction patterns, the ordered structures are characterized by site occupancy factors of 1, while disordered structures have fractional site occupancies representing incoherent averages of diffractions from all possible crystal configurations. Using the Mercury software 43 to generate simulated X-ray powder patterns, it is possible to adjust the occupancy factors to compare the patterns for the ordered structures with those generated for corresponding structures with random occupation of the Ge and As sites. The simulations show small differences in the X-ray patterns. While it is possible that future experiments could detect such differences, the reported experimental powder patterns 3 are in good agreement with the generated X-ray powder patterns of the ordered structures and nearly equally good agreement with the corresponding patterns generated for the disordered structures. This suggests that the structures discussed below are very plausible models, but not the only possibilities.
The search for optimal alloy structures was based on supercells of the Li 3 AsS 4 structure; four and six formula unit supercells were used for the x = 1/4 and x = 1/3 alloys respectively. For each supercell, all unique configurations of Ge substitutions were considered. For each Ge substitution an additional Li was added at a grid of likely positions. QUANTUM ESPRESSO 17 was used to optimize the structures of each of the initial configurations. For each alloy concentration considered, the optimized alloy model was found to have the lowest energy of the configurations considered by 0.2-0.3 eV.
In order to check for additional alloy models, we also sampled a few configurations from larger supercells with the help of symmetry analysis available from Site Occupancy Disorder (SOD) program of Grau-Crespo et al. 48 Although we failed to find alloy structures with lower energy, it is possible that a more exhaustive search may reveal additional possibilities. On the other hand, the alloy models that we did find, appear to make optimal use of the original Li 3 AsS 4 structure and its low energy interstitial sites, as explained in more detail below.  Table AII.
In anticipation of the analysis of Li ion migration in this material, we studied the relative energies of vacancy and interstitial defects. The Pm structure of the 1/4 alloy has 8 distinct Li sites and 6 distinct interstitial sites as shown in Fig. 3a. The corresponding relative vacancy and interstitial energies are listed in Table III. From the simulations based on 16 formula unit supercells, the x site Li ion has the lowest vacancy energy, defined as E = 0. The nearest neighbor  Table III with E = 0 because their relaxed configuration is that of x site vacancy. The vacancy energies for Li ions further from the Ge site are more than 0.5 eV higher in energy. Accordingly, we expect that the population of vacancies on the a 2 , a 3 , b 3 , and b 4 sites to be substantial lower than the population of vacancies on the x sites. However, entropy effects 32 which reduce the free energies, increase the probability of forming these vacancies.
The interstitial energies of sites K and K , both corresponding to the type I interstitial of the pure Li 3 AsS 4 lattice, are 0.06 eV higher in energy than the lowest energy J interstitial site, which is defined as E = 0. The four unique interstitial sites J , L, M, and N all correspond to the the type I I interstitials in pure Li 3 AsS 4 . The lowest energy J site is closest to the plane containing the substitutional Ge atoms, although because of its c axis placement, the interstitial L site interstitial is closer to its nearest Ge site, as noted in Table III.   for this alloy is shown in Fig. 3b. It can be described in terms of a 3 × 1 × 1 supercell of pure Li 3 AsS 4 , with Ge substituting for As in planes perpendicular to the a-axis at the fractional coordinates x = 0 and x = 1/2. The associated 2 extra Li sites are located close by as indicated with the symbols x in the figure. The x site Li's of the alloy occupy positions close to interstitial site of type "I " of the original Li 3 AsS 4 lattice. This structure is 0.27 eV lower in energy than all the other configurations we investigated for this alloy. The lattice constants of this optimized structure are a = 22.77 Å, b = 6.60 Å, and c = 6.14 Å, corresponding to a slight contraction of the Li 3 AsS 4 lattice along the a axis. This structure of the Li 10/3 As 2/3 Ge 1/3 S 4 alloy has the space group Pmn2 1 (No. 31), as does the optimal structure of the Li 3 AsS 4 host lattice. The fractional coordinates of the unique atoms are listed in the Appendix in Table AIII.
In anticipation of the analysis of Li ion migration in this material, we studied the relative energies of vacancy and interstitial defects. The Pmn2 1 structure of the 1/3 alloy has 6 distinct Li sites and 4 distinct interstitial sites as shown in Fig. 3b. The corresponding relative vacancy and interstitial energies are listed in Table IV. From the simulations based on 24 formula unit supercells, the x site Li ion has the lowest vacancy energy, defined as E = 0. As in the case of the 1/4 alloy, the nearest neighbor Li vacancy sites, a 1 , b 1 , and b 2 , are all unstable relative to the x site vacancy. Vacancies at sites further from the Ge substitutions, a 2 and b 3 are approximately 0.5 eV   or higher in energy. As in the case of the 1/4 alloy, we expect that entropy effects will enable the population of vacancies on these to be non negligible. The interstitial sites have relative energies similar to those of the 1/4 alloy. In this case, the lowest energy site is labeled B, defined as E = 0, and has the smallest distance from the plane containing the Ge substitutions.

Densities of states and heats of formation.-
The partial density of states analysis provides qualitative information about the atomic contributions to the occupied and unoccupied states of the materials. Fig. 4 shows the results for the materials in this study. It is interesting that the two alloys with x = 1/4 and x = 1/3 show nearly identical densities of states. The valence band of these materials is primarily composed of the S 3 p states, while the conduction bands are composed of unoccupied As and Ge states, hybridized with S. In general the states associated with Ge lie above their corresponding As counter parts; the bandgap of the alloy is expected to be similar to that of Li 3 AsS 4 .
A more quantitative measure of density functional results is the heats of formation ( H f ) of the materials referenced to the elemental materials in their standard states determined from the total energies of the systems as described by Eq. 1. Also of interest is the decomposition energies ( H D ) for the alloys which can also be estimated from the zero temperature total energy results according to Eq. 2. Both of these energies are listed in Table V. The results show that that Li 4 GeS 4 has the largest magnitude of H f of the materials studied. Small positive values of H D having the order of magnitude of the expected finite temperature and pressure corrections, 32 suggest that at equilibrium the alloys are marginally stable with respect to decomposition into the pure materials.
Ion migration mechanisms.-Using the NEB technique described in the Computational methods section, we examined several possible Li ion migration processes in Li 3 AsS 4 and in the Li 3+x As 1−x Ge x S 4 alloys. The results are summarized in Table VI and the details are described below.
Ion Migration in pure Li 3 AsS 4 .-For defect-free crystalline materials, the energy to form an interstitial vacancy pair, E f contributes to the activation energy for conductivity as described in Eq. 4. For pure Li 3 AsS 4 , we found the lowest formation energy for an interstitialvacancy pair to be E f = 0.86 eV corresponding to a type I interstitial site and a nearby b vacancy site. The formation energy for a type I I interstitial and nearby b vacancy is very similar. The corresponding interstitial-vacancy pair formation energy in γ-Li 3 PS 4 is E f = 0.81 eV due to a type I I interstitial site being occupied by a nearby b vacancy site. 26 Several Li ion vacancy migration paths in Li 3 AsS 4 were investigated according to the diagram shown in Fig. 5a with the resulting NEB energies shown in Fig. 5b. For comparison, the corresponding NEB energies for γ-Li 3 PS 4 are also shown in Fig. 5b. For these paths, the energies for Li vacancy migration in Li 3 AsS 4 are considerably higher than that of the thiophosphate. The migration energies E m are determined as the maximum energy difference along the path which is E m = 0.4 eV for Li 3 AsS 4 and E m = 0.3 eV for γ-Li 3 PS 4 .
Because of the relatively large value of E m for pure Li ion vacancy migration in Li 3 AsS 4 , we were motivated to investigate other diffusion mechanisms such as ion migration involving interstitial Li ion sites. Three different such paths were found as shown in Fig. 6. Two paths with net motion along the b axis are shown in Fig. 6a. The path indicated by red arrows I I 1 → I 1 → I I 4 corresponds to pure interstitial migration. The path indicated by black arrows involves a kick-out process of the I I 1 interstitial ion displacing an "a" host lattice ion into the I 1 interstitial position. Subsequently, the I 1 interstitial Li ion displaces the "b" host lattice ion into the I I 4 interstitial position. The third path, with net motion along the c axis, is shown  in Fig. 6b. This is a purely interstitial migration between neighboring I I type interstitials between sites I I 1 → I I 2 → I I 3 . For this path, the NEB analysis shows a correlated motion of the host lattice Li ion at the neighboring "a" site was found in partial kick-out like process. While this correlated motion contributes to the energetics of the process, it does not seem to directly contribute to ion migration. The corresponding minimum energy paths as determined from the NEB method are shown in Fig. 7. We see that for the paths along the b axis, the kick-out mechanism, with a migration energy E m = 0.34 eV, is more favorable than the pure interstitial mechanism with a migration energy E m = 0.5 eV, and also more favorable than the pure vacancy migration shown in Fig. 5. The pure interstitial migration path along the c axis has a migration energy E m =0.31 eV, which is the most favorable of the paths considered for this material.  Fig. 1. (a) Indicates a migration path with net motion along the b axis, the red arrows indicating the pure interstitial process and the black arrow indicating a related process including the "kick-out" of host lattice ions at each step. (b) Indicates a migration path with net motion along the c axis with a pure interstitial process.  Figure 7. (a) Minimum energy paths as determined from the NEB method for the pure interstitial mechanism along the b axis (red curve) and for the kick-out mechanism (black curve) corresponding to the diagram in Fig. 6a. (b) Minimum energy path as determined by the NEB method for the pure interstitial mechanism along the c axis corresponding to the diagram in Fig. 6b.
Ion migration in Li 13/4 As 3/4 Ge 1/4 S 4 .-For an alloy, it is expected that the role of the interstitial-vacancy pair formation energy E f in contributing to the activation energy as described by Eq. 4 is less important than it is in highly ordered crystals. However, E f is a welldefined quantity within our ordered models for these alloys. For the Li 13/4 As 3/4 Ge 1/4 S 4 alloy, the lowest formation energy was found to be E f = 0.27 eV involving the J type interstitial and its neighboring "x" site vacancy. This value of E f is considerably smaller than that pure Li 3 AsS 4 and its mechanism is a consequence of the extra "x" Li ions introduced by Ge.
For several reasons, it is reasonable to assume that interstitial and interstitiacy ion migration mechanisms, and not pure vacancy mechanisms, are of main importance for ion migration in Li 3+x As 1−x Ge x S 4 alloys. Firstly, as discussed for pure Li 3 AsS 4 , these mechanisms have smaller migration energy barriers in the Li 3 AsS 4 lattice itself. Secondly, as discussed above, the x site Li can contribute to the interstitial population with a relatively small formation energy. Finally, as discussed in the structural analysis section, several potential Li ion vacancy sites in the vicinity of Ge (sites a 1 , b 1 , and b 2 shown in Fig.  3a) are found to be unstable relative to the x site vacancy, limiting the pure vacancy pathways available within this lattice.
As discussed above, for pure Li 3 AsS 4 , the most energetically favorable Li ion migration paths were found to be a kick-out mechanism along b axis and a pure interstitial mechanism along the c axis. In the Li 13/4 As 3/4 Ge 1/4 S 4 alloy, several variations of these paths occur; it is expected that their energy profiles will depend on the location of the paths relative to the planes of Ge substitution.
We studied three representative paths based on a kick-out mechanism with net migration along the b axis, as shown in Fig. 8a. The path L 1 → b 2 → K 2 → a 2 → L 2 has a migration energy of E m = 0.28 eV. It is geometrically equivalent to the path I I 4 → b → I 1 → a → I I 1 with a migration energy of E m = 0.34 eV found in the pure Li 3 AsS 4 structure. A contributing factor to the decrease in E m for this path in the alloy is the alignment of the interstitial energies of the L and K sites due to their proximity to Ge as reported in Table III. A similar path within the 1/4 alloy is represented by the sequence N 1 → a 3 → K 1 → b 3 → N 2 which is found to have a migration energy of E m = 0.31 eV, involving sites further from the Ge substitutions. We also considered the path represented with the sequence J 1 → a 2 → K 1 → b 2 → J 2 . As illustrated in Fig. 9a, the migration energy for this path, E m = 0.52 eV, is so large that this mechanism is unlikely to contribute to the diffusion process. While the geometry of this path is similar to the other two b axis kick-out mechanisms, the hop distances for the first and last steps of the process are 0.2 Å longer. In addition, due to its proximity to the plane of Ge substitutions, the J interstitial site is the most stable of the interstitial sites as shown in Table III. Both of these factors may explain the high migration energy of this path.
For the pure interstitial migration, two different paths with a net migration along the c axis were considered as shown in Fig. 8b. The migration energy was determined from the NEB calculations, as presented in Fig. 9b. These paths are similar to that found in pure Li 3 AsS 4 where it involves the interstitial sites I I 1 → I I 2 → I I 3 . For the path J 3 → L 3 → J 4 , the first hop has a very low migration barrier apparently due to a correlated motion of the a 1 lattice site in a partial "kick-out" like process. For the second hop of this path, there is apparently no correlated motion by the nearby host Li ion of type a 2 and the overall migration barrier for this path is increased to E m = 0.42 eV. The path N 3 → M 1 → N 4 involves the correlated motion of the nearby Li ion at a a 3 site on both hops, resulting in the overall migration energy for this path of E m = 0.29 eV which is slightly lower than the corresponding migration energy of E m = 0.31 eV in pure Li 3 AsS 4 .
Ion migration in Li 10/3 As 2/3 Ge 1/3 S 4 .-For the Li 10/3 As 2/3 Ge 1/3 S 4 alloy, the lowest formation energy was found to be E f = 0.27 eV, involving the B type interstitial and its neighboring x site vacancy, geometrically and energetically similar to the 1/4 alloy.
For modeling b axis migration with the kick-out mechanism, three different paths were considered, as shown in Fig. 10a. The correspond-ing minimum energy paths determined from NEB calculations are presented in Fig. 11a. The path sequence A 1 → a 2 → K 1 → b 2 → A 2 has a migration energy of E m = 0.23 eV which is smaller than E m = 0.28 eV and E m = 0.34 eV for the corresponding migrations in the 1/4 alloy and pure Li 3 AsS 4 , respectively. The path sequence C 1 → a 2 → K 2 → b 3 → C 2 has a migration energy of E m = 0.30 eV which very slightly smaller than E m = 0.31 eV and E m = 0.34 eV for the corresponding migrations in the 1/4 alloy and pure Li 3 AsS 4 , respectively. The third path considered in this study, B 1 → a 2 → K 3 → b 2 → A 3 , has a relatively large migration barrier of E m = 0.39 eV. The first hop of this path involves the B type interstitial site which tends to cause larger migration barriers due to its stability, similar behavior of the J site in the 1/4 alloy. The second hop of this process is identical (within the calculational error) to the For the pure interstitial mechanism along the c axis, two paths were considered as shown in Fig. 10b with the corresponding NEB energy paths shown in Fig. 11b. The path A 1 → B 2 → A 4 has a migration energy of E m = 0.47 eV. As in the case of the 1/4 alloy, the migration energy is large for this case due to the asymmetric correlated motion of nearby Li ions at inequivalent a sites. For the path C 1 → C 2 → C 3 the migration energy is E m = 0.2 eV. As in the similar paths in the 1/4 alloy and the pure Li 3 AsS 4 , the migration involves the correlated partial kick-out of the neighboring Li ion at a a 2 site. In Figure 9. (a) Minimum energy paths as determined from the NEB method for the three distinct kick-out mechanism paths along the b axis shown in Fig. 8a. The line colors of the plot are consistent the arrows in the structural diagram. (b) Minimum energy paths as determined by the NEB method for the pure interstitial mechanism along the c axis corresponding to the diagram in Fig. 8b. The line colors of the plot are consistent the arrows in the structural diagram.
) unless CC License in place (see abstract). ecsdl.org/site/terms_use address. Redistribution subject to ECS terms of use (see 207.241.231.80 Downloaded on 2018-07-19 to IP Figure 10. Ball and stick diagram of the Li 10/3 As 2/3 Ge 1/3 S 4 using the same orientation and ball convention as Fig. 3b. (a) Shows migration paths with net motion along the b axis, with a process including the "kick-out" of host lattice ions at each step, indicated with black, purple, and red arrows. (b) Shows migration paths with net motion along the c axis with a pure interstitial process, indicated with red and black arrows. this case, the resulting process has the lowest migration energy in the study.

Conclusions
As a result of this computational study, we are reasonably confident in our determination that pure Li 3 AsS 4 forms in the Pmn2 1 crystal structure. Our first principles optimizations found this structure to have the lowest energy of those that we studied, and the results are in good agreement with the X-ray diffraction data reported by Sahu et al., 3 as shown in Fig. 2.
The X-ray data of Sahu et al. 3 for the prepared materials with compositions Li 3+x As 1−x Ge x S 4 suggests that they are substitutional alloys based on the Li 3 AsS 4 structure. Our simulations assumed this to be the case, and the results are consistent. Despite the chemical similarities, the Li 3+x As 1−x Ge x S 4 alloy system based on substitutions into the Pmn2 1 structure, is quite different from Li 3+x P 1−x Ge x S 4 which has been shown 7 to undergo several structural transitions within a similar composition range. Presumably, the size similarity of the AsS 4 and GeS 4 building blocks help stabilize the Pmn2 1 structure.
We determined idealized models for the x = 1/4 and x = 1/3 alloys which turned out to be based on 2 × 1 × 1 and 3 × 1 × 1 supercells of Li 3 AsS 4 , respectively. The analysis shows how the Pmn2 1 structure can accommodate efficient ionic transport with a combination of interstitial and host lattice sites. For Li 3 AsS 4 and the two alloy models, the migration of interstitial Li ions along the c axis gives the smallest value of the migration energy E m . The migration of interstitial Li ions along the b axis using a kick-out process also gives a small value for E m . The summary of the calculated values for the migration energies E m as well as the energy E f to form vacancyinterstitial pairs are given in Table VI where they are compared with the experimentally determined activation energies E A . In general, we find E m ≈ E A , suggesting that the prepared materials correspond to the "extrinsic" case described by Eq. 4. The simulations generally show that an important effect of Ge is to adjust the energies of the participating interstitial and vacancy Li ion sites to reduce E m . Another effect of Ge is to stabilize an extra Li ion in the lattice at what we called the "x" site. Undoubtedly, this extra Li contributes to population of mobile interstitial Li ions, although our models cannot explain the details of these processes. The results of our simulations are in general agreement with the trends presented by the experimental work of Sahu et al. 3 One question that has not yet been resolved is the suggestion that the concentration x = 1/3 may be optimal. Our models show that the minimal values of the migration energies follow the trend E m (x = 0) > E m (x = 1/4) > E m (x = 1/3), suggesting that the substitional Ge incrementally reduces the migration barrier and improves the ionic conductivity.
The Li 3+x As 1−x Ge x S 4 alloy system of Li ion conductors is the only system (that we know about) which is based on the Pmn2 1 structure. Several related alloy systems, such as Li 3+x P 1−x Si x S 4 Figure 11. (a) Minimum energy paths as determined from the NEB method for the kick-out mechanism along the b axis corresponding to the diagram in Fig. 10a. The line colors correlate with the arrows in the structural diagram. (b) Minimum energy path as determined by the NEB method for the pure interstitial mechanism along the c axis corresponding to the diagram in Fig. 10b. The line colors correlate with the arrows in the structural diagram.