Substitutional disorder: structure and ion dynamics of the argyrodites Li₆PS₅Cl, Li₆PS₅Br and Li₆PS₅I

Abstract

For the development of safe and long-lasting lithium-ion batteries we need electrolytes with excellent ionic transport properties. Argyrodite-type Li₆PS₅X (X: Cl, Br, I) belongs to a family of such a class of materials offering ionic conductivities, at least if Li₆PS₅Br and Li₆PS₅Cl are considered, in the mS cm⁻¹ range at room temperature. Although already tested as ceramic electrolytes in battery cells, a comprehensive picture about the ion dynamics is still missing. While Li₆PS₅Br and Li₆PS₅Cl show an exceptionally high Li ion conductivity, that of Li₆PS₅I with its polarizable I anions is by some orders of magnitude lower. This astonishing effect has not been satisfactorily understood so far. Studying the ion dynamics over a broad time and length scale is expected to help shed light on this aspect. Here, we used broadband impedance spectroscopy and ⁷Li NMR relaxation measurements and show that very fast local Li ion exchange processes are taking place in all three compounds. Most importantly, the diffusion-induced NMR spin–lattice relaxation in Li₆PS₅I is almost identical to that of its relatives. Considering the substitutional disorder effects in Li₆PS₅X (X = Br, Cl), we conclude that in structurally ordered Li₆PS₅I the important inter-cage jump processes are switched off, hindering the ions from taking part in long-range ion transport.

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1 Introduction

Energy from renewable sources can be stored in various ways. Considering electrochemical storage, Li-based batteries offer high energy densities combined with a high degree of design flexibility.^1–3 In particular, batteries using Li⁺ as an ionic charge carrier are an emerging option to power electric vehicles.^1–7 Such batteries may, however, suffer from flammable fluorinated liquid electrolytes conventionally used in these energy storage systems. Next-generation Li batteries with, e.g., metallic Li anodes and equipped with solid electrolytes have aroused great interest.^8–12 Devices with non-flammable ceramic electrolytes are anticipated to benefit from increased safety as they easily withstand higher temperatures. Thus, such systems are less sensitive to thermal runaway.

To take full advantage of all-solid-state Li (or Na) batteries, electrolytes with exceptionally high ionic conductivities are needed.^8–15 In the past, only a few ceramics were available fulfilling this requirement.^16,17 This lack of suitable materials prevented any commercialization on a large scale.^18,19 Ideal candidates should (i) show room-temperature ionic conductivities with values comparable to those of liquids, (ii) be chemically stable over a sufficiently wide temperature range and (iii) possess a sufficiently high electrochemical stability over a large potential window.^10,11 For battery applications, the conductivity of a ceramic material should reach a value of 10⁻³ S cm⁻¹ at room temperature.^10,14

Over the last years several classes of materials have been presented whose members show very high conductivities. These materials include sulfides, such as Li₇P₃S₁₁ and Li₁₀GeP₂S₁₂, oxides including garnets, such as Li₇La₃Zr₂O₁₂, or phosphates based on the NASICON structure.^{8–15,20–26} Although the ionic bulk conductivities of oxides and phosphates can reach very high values, the materials may suffer from high grain boundary (g.b.) resistances.²⁷ For the softer sulfides the difference between the bulk and g.b. conductivities²⁸ is expected to be much lower. Of course, Li-bearing sulfides or thiophosphates are sensitive to air and moisture, thus requiring careful handling and adequate processing. As an example, the Ge-containing thiophosphate Li₁₀GeP₂S₁₂ (LGPS),¹³ which initiated the renaissance to consider sulfide-based systems as candidates for solid electrolytes, is characterised by an ionic conductivity of 12 mS cm⁻¹. Meanwhile the expensive element Ge has been replaced by cheaper and naturally abundant materials.²⁹ The electrochemical stability of LGPS-based systems needs, however, to be improved.³⁰

Besides LGPS, Deiseroth and co-workers introduced Li-containing argyrodite-type Li₆PS₅X (X = Cl, Br, I).³¹ The crystal structure is shown in Fig. 1; the possible Li exchange processes are illustrated as well. As an example, the ionic conductivity of Li₆PS₅Br and Li₆PS₅Cl is reported to range from 10⁻³ S cm⁻¹ to 10⁻² S cm⁻¹ depending on the preparation technique, which influences the type and number of defects as well as the overall morphology.^32,33 Thus, the ionic conductivity of Li-argyrodites is only slightly smaller than that of the LGPS family. Lower costs make this family of electrolytes highly attractive for application in all-solid-state batteries.^31,34

Fig. 1 (a) Crystal structure of argyrodite-type Li₆PS₅X (X = Cl, Br, I). In a perfect ordered structure, a cubic lattice is formed with PS₄³⁻ tetrahedra and Li ions being arranged octahedrally. Some of the S²⁻ (4d) anions are placed in the middle of the Li octahedra; the anions on 4a form a face centred cubic lattice. (b) The two different lithium positions (24g, 48h) form an octahedral arrangement in which three different jump processes can occur: (i) strictly localised jumps between the Li positions 48h–24g–48h′; (ii) intracage jumps (48h–48h′′) and (iii) so-called intercage jumps 48h₁–48h₂ between two neighbouring cages 1 and 2. The latter jump process forms interconnected diffusion pathways being responsible for 3D long-range ion transport. Site disorder on the anion positions 4a and 4d, which is absent for Li₆PS₅I, is expected to influence the Li ion diffusivity significantly.

In 2008 Deiseroth et al. reported the solid-state synthesis of Li₆PS₅X (X = Cl, Br, I) and characterised the local and long-range structure by X-ray diffraction and NMR.³¹ Three years later Rao et al. discussed the synthesis of the lithium argyrodite via ball milling.³⁵ Rayavarapu et al. showed that the disorder in both the lithium distribution over the available cation sites and the disorder in the S²⁻/Cl⁻ and S²⁻/Br⁻ sublattices promotes lithium ion conductivity.³⁶ This view has further been supported by bond-valence calculations.³⁷

Over the last years a number of studies have been carried out to measure the ionic conductivities of the Li₆PS₅X family by using standard impedance measurements.^31–40 The overall goal, not only for Li₆PS₅X, is to relate macroscopic application relevant properties to crystal chemical and morphological properties. So far, the results reported have rarely been related to the length-scale specific properties of the measuring technique applied. The ionic conductivities for Li₆PS₅X vary from 10⁻⁷ S cm⁻¹ to 10⁻² S cm⁻¹, while the highest values are found for X = Cl, Br. The corresponding activation energies reported, either experimentally probed or calculated, take values from 0.11 eV to 0.57 eV.^31–40 Astonishingly, the I-compound Li₆PS₅I shows the lowest conductivity. According to the strategy of substitutional disorder, size mismatch and polarizability of the halogen anion one could have expected the opposite trend. Replacing S by a larger and more polarizable anion such as I, as compared to the small and unpolarizable Li cation, is expected to cause lattice distortions which on their part lead to a broader distribution of slightly inequivalent Li sites in the Li sublattice. Without any strong site preference of the Li ions they are expected to jump quickly between the local minima of such a heterogeneous, i.e., non-uniformly shaped, potential landscape. Such a situation, sometimes also connected to geometric frustration, is very similar to F diffusion in the recently investigated system (Ba,Ca)F₂,^41,42 Li diffusion in Li₇La₃Zr₂O₁₂ oxides⁴³ and Na diffusion in some closo-borates.⁴⁴ While this situation might be the origin for the high conductivity in Li₆PS₅Br and Li₆PS₅Cl, for the I analogue it seems to incompletely describe the real situation as for Li₆PS₅I relatively low ion conductivities are reported.

In an elegant way, Kraft et al.³⁴ tried to establish a connection between the stiffness (or polarizability) of the anion sublattice in Li₆PS₅X and the macroscopic ionic transport properties. They used impedance spectroscopy and ultrasonic speed of sound measurements to collect information on activation energies and the pre-factors governing the underlying Arrhenius equation. They reported that soft lattices, as is the case for Li₆PS₅I, lead to smaller Arrhenius pre-factors than obtained for Li₆PS₅Cl and Li₆PS₅Br. This decrease in pre-factor, which might originate from a change of the migration entropy or attempt frequencies, has been used to explain the low (overall) ion conductivity in Li₆PS₅I.

Here, we employed impedance and NMR spectroscopy to shed more light on the effect seen for Li₆PS₅I. The important structural differences between the three argyrodites have to be taken into account to explain the significant changes in ionic conductivities, pre-factors and activation energies. For this purpose we have not only looked at the monosubstituted compounds like Li₆PS₅X, but also at samples with two or three different anions such as Li₆PS₅Cl_0.5Br_0.5 or Li₆PS₅Cl_1/3Br_1/3I_1/3. The non-substituted compound Li₇PS₆ served as a reference material. Importantly, complementary techniques which are sensitive to ion dynamics on different length-scales and time-scales are used to fully characterise the transport properties. We will show that, on a short-range length scale, the Li ions in Li₆PS₅I are as rapid as in the Br and Cl analogues. Obviously, the ordered S²⁻/I⁻ sublattice, as compared to the disordered situation for Li₆PS₅Cl and Li₆PS₅Br, is responsible for a more regular potential landscape with a higher degree of site preference for the Li ions. As a consequence, we assume that intercage jump processes in Li₆PS₅I occur less frequently. These processes are, however, needed to establish rapid long-range ion dynamics in the iodide.

2 Experimental

2.1 Sample preparation through a solid state reaction

The Li argyrodites Li₆PS₅X, with the compositions shown in Fig. 2b, have been prepared by mixing stoichiometric amounts of Li₂S (Sigma-Aldrich ≥99.98%), P₂S₅ (Honeywell Fluka ≤99%) and LiCl (Sigma-Aldrich, anhydrous, 99%), LiBr (Sigma-Aldrich, anhydrous, 99.98%) and LiI (Sigma-Aldrich, 99.9%). The starting compounds were mechanically milled using a high-energy planetary ball mill (Fritsch Pulverisette 7 Premium line) at a rotation speed of 400 rpm for 4 hours (15 min milling; 15 min pause). Milling was carried out using ZrO₂ beakers (45 ml) filled with 180 balls (ZrO₂, 5 mm in diameter). Afterwards the mixtures were uniaxially pressed (0.4 tons) into pellets with a diameter of 5 mm, filled into quartz ampoules and sealed under a vacuum. The ampules were preheated at 573 K under a dynamic vacuum to remove traces of water. The reaction was carried out at 823 K (heating rate: 5 K min⁻¹, duration 7 days; cooling rate 10 K min⁻¹). All preparation steps were strictly carried out in an Ar gas atmosphere (H₂O < 1 ppm, O₂ < 1 ppm).

Fig. 2 (a) X-ray powder diffraction patterns of the Li₆PS₅X series investigated within this study (X = Cl, Cl_0.5Br_0.5, Br, Br_0.5I_0.5, I and Cl_0.33Br_0.33I_0.33). The XRPD patterns were recorded after the final annealing step at room temperature. In some cases, little, almost negligible, amounts of Li₃PO₄ and LiX are visible; the amount of LiX does not exceed 1.5 wt%. Only for Li₆PS₅Cl ca. 3.3 wt% Li₃PO₄ is seen, see the arrows; in all other cases its amount is negligible or less than 1.5 wt%, see the ESI† for Rietveld refinements (Fig. S1 and S2). (b) Lattice parameter of Li₆PS₅X obtained from Rietveld refinements of the patterns shown in (a). The lattice parameter increases continuously with the radius of X.

2.2 X-ray powder diffraction (XRPD)

The samples were analyzed by X-ray diffraction using a Bruker D8 Advance diffractometer (Bragg Brentano geometry, Cu K_α radiation). The patterns, shown in Fig. 2, were recorded with a step size of 0.02° (measuring time 2 s) in the 2θ-range 20° to 100°. Rietveld refinement, see Fig. 3 and the ESI† (Fig. S1 and S2), was carried out by using X'PertHighScorePlus (Panalytical).

Fig. 3 Rietveld analysis of the XRPD pattern of Li₆PS₅Br recorded at 293 K (R_Bragg = 4.633%). In this plot the Bragg positions as well as calculated and observed profiles are depicted. The analysis reveals a phase pure sample. The R_w profile is 10.84% and the GoF amounts to 4.82; see the ESI† for further explanations.

2.3 Impedance spectroscopy

To measure the impedance responses of the pellets, Au electrodes were applied. We used a LEICA EM SCD 050 sputter coater to equip the samples with a thin Au layer of 100 nm thickness. Impedance spectra were recorded with a Novocontrol Concept 80 broadband dielectric spectrometer; the frequency range covered ranged from ν = 10 mHz to 10 MHz. The temperature T was varied from 173 K to 453 K in steps of 20 K; the temperature in the sample holder was controlled by a QUATRO cryosystem (Novocontrol) that uses freshly evaporated nitrogen and a heater to adjust the temperature with an accuracy of ±0.5 K. The measurements were carried out in a dry nitrogen atmosphere to avoid any contamination with water and oxygen.

2.4 ⁷Li static NMR measurements

For the time-domain NMR measurements the samples Li₆PS₅Cl, Li₆PS₅Br and Li₆PS₅I were sealed in Duran© glass tubes (ca. 4 cm in length and 3 mm in diameter). They were kept under a vacuum to protect them from any reaction with humidity or traces of oxygen. We used ⁷Li NMR line shape measurements and spin–lattice relaxation (SLR) experiments to collect information about Li activation energies and jump rates. Longitudinal NMR SLR rates (1/T₁) as well as spin-lock rates (1/T_1ρ) were measured by employing a Bruker Avance III spectrometer connected to a shimmed cryomagnet with a nominal magnetic field of 7 T. This field corresponds to a ⁷Li Larmor frequency of ω₀/2π = 116 MHz. For the measurements at temperatures T ranging from 173 K to 433 K a ceramic high-temperature probe (Bruker Biospin) was used. At a power level of 200 W the π/2 pulse length ranged from 2.7 to 3.2 μs slightly depending on temperature. A Eurotherm temperature controller in combination with a type T thermocouple was used to control and to monitor the temperature in the sample chamber. Low temperature measurements down to T = 124 K were performed with a cryo probe (Bruker) that uses freshly evaporated nitrogen to reach temperatures below ambient. The π/2 pulse length of the cryo probe ranged from 3.0 to 4.3 μs at a power level of 95 W. It is equipped with a LakeShore 331 element equipped with two Cernox temperature sensors.

⁷Li NMR SLR rates 1/T₁ ≡ R₁ in the laboratory frame were acquired with the well-known saturation recovery pulse sequence.^45,46 This sequence uses a comb of closely spaced π/2 pulses to destroy any longitudinal magnetization M_z. The subsequent recovery of M_z was detected as a function of waiting time t_d with a π/2 reading pulse: 10 × π/2 − t_d − π/2 − acquisition.¹⁵ To construct the magnetization transients M_z(t_d), we plotted the area under the free induction decays (FIDs) vs. t_d. Up to 16 FIDs were accumulated for each waiting time.

The transients M_z(t_d) were parameterised with stretched exponentials, M_z(t_d) ∝ 1 − exp(−(t_d/T₁)^γ), to extract the SLR rates R₁. Additionally, rotating-frame ⁷Li NMR SLRρ rates 1/T_1ρ ≡ R₁ were measured by means of the spin-lock technique: π/2 − p_lock acquisition. We used a locking frequency ω₁/2π of 20 kHz.^47,48 The duration t_lock of the locking pulse p_lock was varied between 22 μs and 460 ms. To ensure full longitudinal relaxation between each spin-lock scan the recycle delay was set to at least 5 × T₁. Again, the R_1ρ rates were obtained by analysing the resulting transients M_ρ(t_lock) with stretched exponentials of the form M_ρ(t_lock) ∝ exp(−(t_lock/T_1ρ)^κ). While the stretching exponents γ varied from 1 to 0.9, the exponents κ range from 0.4 to 1, depending on the temperature. Some of the T_1ρ transients showed bi-exponential behaviour and were analyzed with a sum of two stretched exponential functions.

2.5 Magic angle spinning (MAS) NMR

⁶Li (73.6 MHz), ³¹P (202.4 MHz), ¹²⁷I (100.1 MHz), ⁷⁹Br (125.3 MHz) and ³⁵Cl (49.0 MHz) NMR spectra under magic angle spinning (MAS) conditions were recorded with a 500 MHz Avance spectrometer (Bruker). The high-resolution MAS NMR spectra were measured at a rotation speed of 25 kHz using 2.5 mm rotors with an ambient bearing gas temperature. Spectra were recorded with a single pulse sequence. We accumulated up to 128 scans for one spectrum. LiCl, KBr, LiI and 85% H₃PO₄ served as ref. 31 to determine chemical shifts δ_iso. In order to detect also very slow diffusion processes and to obtain quantitative MAS NMR spectra we varied the recycle delay from 1 s to 180 s. Sufficiently long relaxation delays guarantee full longitudinal relaxation of all spectral components. Further information on pulse lengths, temperature effects and delay times are given in the ESI,† see Table S1.

3 Results and characterization

3.1 Sample characterization by X-ray powder diffraction

To study the ion dynamics in argyrodite-type Li₆PS₅X we used a solid-state synthesis approach to prepare samples with the following compositions: X = Cl, Cl_0.75Br_0.25, Cl_0.5Br_0.5, Cl_0.25Br_0.75, Br, Br_0.75I_0.25, Br_0.5I_0.5, Br_0.25I_0.75, I and X = Cl_0.33Br_0.33I_0.33, respectively. Through replacing S²⁻ with X⁻ we change the average lattice constant of the crystal structure, the polarisability of the anion lattice and the degree of anion disorder. The relative density of our samples, estimated from the sintered pellets used for impedance spectroscopy, reached values ranging from 92% to 98%. Argyrodite-type Li₆PS₅X is expected to crystallise with cubic F43̄m face centered (fc) symmetry; the crystal structure is depicted in Fig. 1.

In Fig. 2a the XRPD patterns of all samples are shown including a reference pattern taken from the literature (entry no. 418490 in the inorganic crystal structure database (ICSD)). The positions and intensities of the reflections obtained for the different patterns of Li₆PS₅X including Li₇PS₆ match very well with what is expected from the literature. Humps at low diffraction angles originate from the mercapto foil used to protect the samples from reaction with air during the measurements.

To characterise the samples in detail, Rietveld refinements were carried out. As an example, the result of the analysis is shown for Li₆PS₅Br in Fig. 3. The Li₆PS₅Br sample is phase pure and crystallises with cubic symmetry (space group F43̄m); the lattice parameter turned out to be a = 9.986 Å. The same symmetry is found for the other samples; the amount of side phases or impurities turned out to be extremely low, see the ESI.† When going from Li₆PS₅Cl to Li₆PS₅I the lattice parameter a increases from a = 9.857 Å to a = 10.145 Å. Lattice expansion of the unit cell is expected because the radius r of the anion increases in the order r_Cl < r_Br < r_I. Here, an increasing unit cell volume stabilises the lithium ions in the transition state 24g; the change in lattice constant a seems to also affect the Li occupancies in the Li₆PS₅X series as has been shown by neutron diffraction.³⁴

Recent neutron and X-ray synchrotron diffraction studies have shown that in fc cubic argyrodite-type Li₆PS₅Cl and Li₆PS₅Br the S²⁻ and X anions can occupy three different crystallographic positions. Whereas the 16e site is fully occupied by S²⁻ anions, the sites 4a and 4d are shared by the S²⁻ and X anions. For Li₆PS₅Cl Kraft et al. found that 38.5% of the 4a site and 61.5% of the 4d site is occupied by Cl⁻.³⁴ The rest is filled by S²⁻ anions; Li⁺ only occupies the 48h sites. For Li₆PS₅Br Rietveld analysis yielded occupancies of 77.9% (4a) and 22.1% (4d) with Li⁺ distributed over the 48h (44.1%) and 24g (11.9%) sites. Thus, the highest degree of anion disorder is found for Li₆PS₅Cl, while cation disorder for the samples rich in Br. This finding is in contrast to what Rietveld refinement resulted in for Li₆PS₅I. While Li disorder is also present for the split-site (48h (39.1%) and 24g (21.9%)), in Li₆PS₅I the I anions solely occupy the 4a sites. The 4d sites are fully occupied by S²⁻. Deiseroth and co-workers also reported on the same order/disorder effects earlier.^31,40,49

The larger difference in ionic radii between I⁻ and S²⁻ (r_I⁻ = 216 pm vs. r_S²⁻ = 184 pm, as compared to r_Cl⁻ = 181 pm and r_Br⁻ = 195 pm) might lead to this site preference for the anions. The anion sublattice in Li₆PS₅I is, thus, structurally ordered as compared to Li₆PS₅Cl (and Li₆PS₅Br); simultaneously, cation disorder shows up for Li₆PS₅Br and Li₆PS₅I.^36,37,50 The compounds with mixed halogen compositions agree well with the trend in S²⁻/X occupancy and cation disorder.³⁴ As will be discussed below, anion ordering seems to have an important impact on long-range ion diffusion in the I compound.

The 24g site represents an intermediate state used by the Li ions on 48h to diffuse through. As a result of these two lithium positions three different jump processes can occur; they are explicitly shown in Fig. 1b. (i) A strongly localised process is given by 48h–24h–48h′ jumps. Intracage jumps are possible within the octahedral cage when the Li ions jump between the 48h sites: (ii) 48h–48h′′. (iii) Long-range ion dynamics needs 48h₁–48h₂ intercage jump processes between two different cages 1 and 2. The latter jump process is expected to be strongly affected by the anion site disorder of X and S²⁻. We expect that a high degree of substitutional disorder will have a significant impact on the associated jump rate, which characterises exchange processes between the cages.

3.2 Magic angle spinning NMR

To investigate local structures of the argyrodites, we recorded ³¹P (spin-quantum number I = 1/2) MAS NMR spectra of Li₆PS₅X (X = Cl, Br, I) at a spinning speed of 25 kHz. The spectra are shown in Fig. 4a. In line with the report of Deiseroth and co-workers,³¹ the ³¹P MAS NMR resonance of Li₆PS₅I shows a very sharp line centered at δ_iso = 96.3 ppm pointing to an ordered anion framework with the I anions non-randomly occupying selected crystal sites. Small shoulders near the main signal might point to defect sites in Li₆PS₅I.

Fig. 4 (a) ³¹P MAS NMR spectra of Li₆PS₅I, Li₆PS₅Br and Li₆PS₅Cl recorded at 202.4 MHz and a spinning frequency of 25 kHz. The spectra have been referenced to 85% H₃PO₄ and were recorded at an ambient bearing gas temperature.³¹ (b) ⁶Li MAS NMR spectra of Li₆PS₅Cl, Li₆PS₅Br and Li₆PS₅I recorded at 73.6 MHz and a spinning speed of 25 kHz. All spectra have been referenced to solid LiCH₃COO. Values given in ppm indicate the isotropic chemical shifts.

The ³¹P MAS NMR spectrum of Li₆PS₅I is in stark contrast to the situation that is met for Li₆PS₅Cl, for which we expect the influence of strong S/Cl disorder on the line shape, see above. Indeed, its very broad line, which can hardly be resolved, reveals a wide distribution of chemical shifts and strong spin–spin interactions. Chemical shifts of lines constituting the overall line range from 85 to 81 ppm. Therefore, ³¹P MAS NMR clearly reveals anion disorder in Li₆PS₅Cl. The ³¹P MAS NMR line of Li₆PS₅Br takes an intermediate position. It is composed of narrower lines than those seen for Li₆PS₅Cl; nevertheless, line broadening, as compared to Li₆PS₅I, reveals disorder in the S/Br sublattice. At least the spectrum is composed of three distinct lines located at 93.9 ppm, 92.7 ppm and 91.1 ppm, respectively. The same increase in substitutional disorder might be seen if we consider the corresponding ¹²⁷I, ⁷⁹Br and ³⁵Cl MAS NMR spectra of the three compounds; the corresponding spectra of the three quadrupole nuclei are shown in the ESI,† Fig. S3. Clearly, because of the large quadrupole moment of I (|Q(¹²⁷I)| = 0.721b) the full spectrum including its spinning side bands is rather broad. The isotropic signal, however, is much smaller than that for Li₆PS₅Cl and Li₆PS₅Br with |Q(³⁵Cl)| = 0.855b and |Q(⁷⁹Cl)| = 0.330b. Although exposed to less second order quadrupole effects, the line of Li₆PS₅Cl is as broad as that of the Br compound. Site disorder in Li₆PS₅Cl might explain this additional broadening.

The ordered anion framework seen for Li₆PS₅I by ³¹P MAS NMR, synchrotron X-ray diffraction and neutron diffraction³⁴ is fully consistent with results from ⁷Li SLR NMR on the ion dynamics in the I compound. As we will show and discuss below, an almost symmetric diffusion-induced ⁷Li NMR rate peak describes longitudinal spin–lattice relaxation in Li₆PS₅I; Li⁺ disorder on the split-site does not influence this symmetry. This observation mainly points to the absence of pronounced site disorder in the anion sublattice. Structural anion disorder is, thus, expected to have a significant impact on both the Li ion diffusion and ionic conductivity in Li₆PS₅X.

For the sake of completeness, we also recorded ⁶Li (I = 1) MAS NMR spectra at an ambient bearing gas temperature, these spectra are shown in Fig. 4b. They have been referenced to LiCH₃COO. For Li₆PS₅Br and Li₆PS₅Cl we observe a slightly asymmetric line with chemical shifts of δ_iso = 1.6 ppm and δ_iso = 1.3 ppm, respectively. The line of Li₆PS₅I turned out to be more symmetric in shape and is broader than the other lines. As Li₆PS₅Br and Li₆PS₅Cl show rapid Li ion long-range translational dynamics their lines represent already coalesced spectra at the measurement temperature of 307 K, see the ESI.† This observation is in perfect agreement with ⁷Li NMR motional line narrowing (see below), which has reached its extreme limit already well below ambient temperature.

In contrast, complete line narrowing of the static ⁷Li NMR line of Li₆PS₅I is shifted toward much higher temperature, revealing significantly slower long-range ion dynamics (vide infra). In addition, the absence of effective Li ion dynamics in Li₆PS₅I, which would be able to also average (small) second order quadrupolar interactions that the spin-1 nucleus ⁶Li is exposed to, might further increase the width of the NMR line of Li₆PS₅I as compared to the samples with X = Cl or Br. The increased anisotropy seen for Li₆PS₅Cl is in line with the degree of site disorder in Li₆PS₅Cl and Li₆PS₅Br.

Interestingly, according to the halogen anion introduced into the argyrodite struture the isotropic chemical shift values δ_iso of the ⁶Li MAS NMR spectra increase from Li₆PS₅Br to Li₆PS₅I and further to Li₆PS₅Cl. Thus, the paramagnetic component of the NMR line decreases; obviously, the different halogen anions change the electron density distribution in the direct neighborhood of the Li spins with Li₆PS₅Br having the highest density.

3.3 Ion dynamics as seen by conductivity spectroscopy

We used broadband impedance spectroscopy^51–53 covering frequencies from the mHz to the MHz region to study the ion dynamics in Li₆PS₅X to see the change in the overall ionic transport properties with the X substitution. As an example, in Fig. 5a the real part σ′ of the complex conductivity [small sigma, Greek, circumflex] of Li₆PS₅Br is plotted vs. the frequency ν.

Fig. 5 (a) Conductivity isotherms (10 mHz to 10 MHz) of Li₆PS₅Br at temperatures ranging from 173 K to 453 K; isotherms have been recorded in steps of 20 K. (b) Change of the DC conductivity of Li₆PS₅X as a function of inverse temperature. The highest conductivity is found for Li₆PS₅Cl and the lowest for Li₆PS₅I. Data points and the Arrhenius line referring to Li₇PS₆ have an offset of −2 on the log scale for the sake of clarity.

At low frequencies and high temperatures T the isotherms σ′(ν) reveal electrode polarization, which results from the accumulation of ions at the surface of the ion blocking electrodes applied.^52–54 Towards higher frequencies or at sufficiently low T, distinct direct current (DC) plateaus show up, which reflect long-range ion transport. The corresponding σ_DC value is governed by both bulk and grain boundary contributions. For the bromide sample, Li₆PS₅Br, log₁₀(σ_DC)T follows Arrhenius behaviour

σ_DCT = σ₀ exp(−E_a,DC/(k_BT)) (1)

with an activation energy, E_a,DC, as low as 0.296 eV. The corresponding Arrhenius line is shown in Fig. 5b. Almost the same behaviour is seen for Li₆PS₅Cl_0.5Br_0.5. Going back to the underlying isotherms σ′(ν) we clearly observe dispersive regions at very low T and high frequencies ν. These regions can be approximated by a Jonscher-type power law, σ′ = σ_DC + Aν^p, where A is the alternating current coefficient and p represents the power law exponent. Because of the very high ionic conductivity of Li₆PS₅Br, this behaviour, influenced by short-range ion dynamics including forward–backward jump processes, is only seen at temperatures below 193 K.^52,54 Here, p turned out to be ca. 0.75; such a value is expected for 3D correlated motion.⁵⁵ Similar isotherms were observed for the other compositions investigated.

For comparison, in Fig. 5b the temperature dependence of log₁₀(σ_DC)T of the other samples is presented. Li₆PS₅I and Li₇PS₆ are characterised by the lowest conductivities and the highest activation energies E_a,DC of 0.47 eV and 0.37 eV. At approximately 250 K we notice a slight deviation from the Arrhenius line determining log₁₀(σ_DC)T at higher T. This behaviour cannot be related to the phase transition of Li₆PS₅I;^31,39,40,49 the I compound transforms into a low-T phase at T ≈ 160 K, see below. Such a slight deviation is also seen for Li₆PS₅Cl.

In Fig. 6a the changes in activation energies, ionic conductivities and Arrhenius pre-factors (σ₀, see eqn (1)) are illustrated. In all cases, the replacement of sulfur anions in Li₇PS₆ by heavier halide anions leads to a decrease in activation energies. Except for Li₆PS₅Cl and Li₆PS₅Cl_0.75Br_0.25 the Arrhenius pre-factor σ₀ also decreases. This feature, which is seen when Li₆PS₅Br is compared with Li₇PS₆ (or Li₆PS₅Cl), has been interpreted in terms of the Meyer–Neldel⁵⁶ rule and related to a reduction in Debye frequencies for soft lattices created by substitution of Br and I for S.³⁴ If we compare our results for Li₆PS₅Cl with respect to the conductivity behaviour of Li₇PS₆ we witness both a decrease in E_a,DC from 0.536 eV to 0.396 eV and an increase in σ₀ by one order of magnitude. This favorable combination of E_a,DC and σ₀ results in a room temperature ionic conductivity of σ_DC = 3.8 mS cm⁻¹. For Li₆PS₅Br, with its more polarisable anions on the other hand, the low activation energy is compensated by a pre-factor two orders of magnitude lower than that for Li₆PS₅Cl. Considering high temperatures this feature causes σ_DC to adopt values lower than those of Li₆PS₅Cl.

Fig. 6 Activation energies E_a,DC of the argyrodite samples prepared. E_a,DC, determined with the help of conductivity isotherms, characterises the overall ion transport, which is affected by both bulk and g.b. contributions. A minimum is found for Li₆PS₅Br. The dependence of the corresponding Arrhenius pre-factor on anion substitution is also shown. For the sake of completeness, we also included activation energies that, most likely, describe bulk ion dynamics. These have been extracted either (i) from conductivity isotherms (Li₆PS₅Cl), (ii) from modulus spectra or (iii) from variable-temperature resistivity measurements carried out at fixed frequencies (see below). In the upper graph overall ionic conductivities at room temperature are shown. The highest conductivity of 3 × 10⁻³ S cm⁻¹ has been observed for Li₆PS₅Cl, whereas Li₆PS₅I reveals the lowest overall conductivity of only 10⁻⁶.

Interestingly, the ionic conductivity of ordered Li₆PS₅I is even lower than that of the non-substituted compound Li₇PS₆; this decrease, despite the higher activation energy found for Li₇PS₆, can also be explained by the relatively low value for σ₀. The pre-factor is proportional to the attempt frequencies ν_a, the number of effective charge carriers n_Li and an entropy term.⁴² Besides the effect of ν_a, trapping effects in Li₆PS₅I, because of a very low intercage jump rate, might reduce N for this sample. If we assume similar ν_a values for Li₆PS₅Cl and Li₆PS₅Br, entropy effects could serve as an explanation for the change in σ₀. Especially for the highly disordered Li₆PS₅Cl sample⁵⁰ the migration entropy might take a decisive role in influencing the ion dynamics. For the ordered Li₆PS₅I compound this contribution might be absent.

When going from Li₆PS₅Br to Li₆PS₅I we see that, despite the introduction of polarisable iodine anions, the pre-factor remains almost the same, A = log(σ₀/(S cm⁻¹ K)) = 4.9 for Li₆PS₅Br and A = 4.6 for Li₆PS₅I, but E_a,DC increases from 0.296 eV to 0.47 eV. From this point of view, although the lattice gets softer in the case of I, Kraft et al. mentioned³⁴ that Meyer Neldel seems to be not fully applicable to describe the situation in Li₆PS₅I. One could have expected the more polarisable I anions to allow the Li ions to squeeze through smaller voids.⁵⁰ Obviously, this concept does not work for Li₆PS₅Br and Li₆PS₅I, at least if we regard the (overall) averaged long-range ion transport process as probed by σ_DC measurements.

The ionic conductivities of the mixed compounds containing I anions, viz. Li₆PS₅Br_0.5I_0.5 and Li₆PS₅Cl_1/3Br_1/3I_1/3, do not vary much. Obviously, the introduction of I plays the crucial role to lower σ_DC. The Cl anions do not affect the ionic transport in these high-entropy compounds in a positive way, at least for compounds containing two anions greatly differing in ionic radius (181 pm vs. 216 pm). To take a closer look at the exceptional behaviour of Li₆PS₅Cl with its high degree of anion disorder we tried to separate the bulk and grain boundary contributions of this sample. Moreover, we used complex resistivity measurements to further characterise the ion dynamics.

The σ′(ν) isotherms of Li₆PS₅Cl are shown in Fig. 7a. A careful look at the DC plateau region shows that it contains a small step. Thus, it is composed of actually two plateaus. Below 250 K, the corresponding activation energies turned out to be 0.310 eV and 0.305 eV, see Fig. 7b. The two-step behaviour of the σ′(ν) isotherms of Li₆PS₅Cl are also seen if we plot the real part ε′ of the complex permittivity vs. ν (Fig. 7b). We used the equation for a parallel-plate capacitor to estimate the corresponding capacitance C of the two steps:

Here ε₀ represents the electric field constant (8.854 × 10⁻¹² F m⁻¹), and A is the area and d the thickness of the sample. While the first DC plateau at high frequencies is characterised by C₁ = 4.1 pF, for the one at lower ν we found C₂ = 1.2 nF. Therefore, the plateau associated with C₁ in the pF range represents the bulk response, whereas the response at lower frequencies is governed by grain boundary contributions for which capacitances typically in the nF regime are expected.⁵⁷ The estimated capacitances are in good agreement with those that can be inferred from complex plane plots (Fig. 7c). At 173 K the corresponding Nyquist plot is dominated by two depressed semicircles; electrode polarization shows up at low frequencies. If we calculate C via ω_el = 1/RC, where ω_el = 2πν_el is the electric relaxation frequency and R the resistance at the intercept of the curve with the Z′ axis,⁵⁷ we obtain C₁ = 4.4 pF and C₂ = 6.8 nF, being in good agreement with the estimation presented above. Only for Li₆PS₅Cl we managed to separate the small difference between the bulk and g.b. contributions. Also for Li₆PS₅I this difference is expected to be very small.

Fig. 7 (a) Conductivity isotherms of Li₆PS₅Cl recorded at temperatures ranging from 173 K up to 393 K. Jonscher-type power laws are used to parameterise the dispersive regions seen at the lowest temperatures. At sufficiently low T we notice a step in the DC plateaus that is attributed to contributions from grain boundaries and the bulk regions. (b) Real part of the permittivity (and loss factor (173 K)) plotted as a function of frequency. Disregarding the polarisation regime, ε′ also reveals a two-step frequency decay characterised by the following capacitances: 4.1 × 10⁻¹² F and 1.2 × 10⁻⁹ F. Dots show the change of the loss factor tan δ passing through two maxima. (c) Complex plane plot (Nyquist plot) of the real part Z′ vs. the imaginary part Z′′ of the complex impedance Ẑ of a Li₆PS₅Cl pellet (173 K). Two depressed semicircles are seen reflecting the bulk (4.4 × 10⁻¹² F) and grain boundary response (6.8 × 10⁻⁹ F).

For comparison, the two contributions in the electrical relaxation of Li₆PS₅Cl are also seen in the loss factor tan δ = ε′′/ε′ when plotted as a function of frequency ν (see also Fig. 7b). The maxima in tan δ, showing up at 15.36 Hz (g.b.) and 4.21 kHz (bulk), correspond to the plateaus or inflection points of ε′ν, see also Fig. 8. According to [small sigma, Greek, circumflex] = iωε₀[small epsilon, Greek, circumflex] = iωε₀ε′ + ωε₀ε′′ the maxima in tan δ produce two minima if the frequency dependence of the imaginary part σ′′(∝ε′) is analyzed (Fig. 8).

Fig. 8 Comparison of the frequency dependence of tan δ, ε′ and σ′′ of Li₆PS₅Cl measured at 173 K. Plateaus (or inflection points) in ε′ result in minima in σ′′ and maxima in tan δ, respectively.

To further characterise ion hopping in Li₆PS₅X, especially if we consider samples with lower conductivity than Li₆PS₅Cl and Li₆PS₅Br, we used the electric modulus representation^58,59 to take a look at electrical relaxation frequencies τ_M′′⁻¹ = ν_max. As an example, in Fig. 9 for Li₆PS₅Br_0.5I_0.5 the imaginary part M′′ of the complex modulus is plotted vs. ν. Since the amplitude of M′′ is proportional to 1/C, bulk processes are mainly seen. For samples with very high conductivities, such as Li₆PS₅Cl, the peaks are shifted to too high frequencies and are, thus, no longer fully visible. Activation energies E_a,M′′ from our M′′(ν) analysis are also included in Fig. 6. For comparison, the bulk activation energy deduced from Fig. 7 is included, too. They reveal that the activation energies, which we ascribe to bulk properties, do not change much if we consider compounds up to the composition Li₆PS₅Br_0.5I_0.5. This behaviour changes if we increase the I concentration. Importantly, for the compounds rich in I, especially if we consider Li₆PS₅I, the observation E_a,M′′ < E_a,DC suggests that the charge carrier concentration increases with temperature. For Li₆PS₅I the difference E_a,DC − E_a,M′′ turned out to be relatively high, viz. 0.121 eV (see the vertical arrow in Fig. 6).

Fig. 9 (a) Frequency dependence of the imaginary part of the modulus, M′′, of Li₆PS₅Br_0.50I_0.50 recorded from 173 K to 253 K in steps of 20 K. The lines are to guide the eye. The temperature dependence of the rate τ_M⁻¹ is shown in (b). For comparison, the results for Li₆PS₅Br_0.25I_0.75, Li₆PS₅Br_0.75I_0.25 and Li₆PS₅I are also included. The lower part of the graph shows σ_DCT(1/T); the values given represent activation energies. For the sake of clarity, data of Li₆PS₅Br_0.75I_0.25 (solid line, grey) have been plotted using an offset of +1 on the log scale.

Additionally, after we took a look at the overall and bulk ion dynamics in Li₆PS₅X we used complex resistivity measurements, [small rho, Greek, circumflex] = 1/[small sigma, Greek, circumflex] = M̂/(iωε₀), carried out at constant frequency but varying temperature, to answer the question of how the bulk ion dynamics depends on the time-scale and length-scale that the technique applied is sensitive to. In Fig. 10a the temperature dependence of the real part ρ′ = M′′/ω of the complex resistivity [small rho, Greek, circumflex] is shown for Li₆PS₅Br. The peaks in Fig. 10a have been recorded for two different frequencies viz. 1 MHz and 10 MHz.

Fig. 10 (a) Real part of the complex resistivity ρ′ of Li₆PS₅Br, measured at a fixed frequency (1 MHz and 10 MHz), vs. 1000/T. (b) ρ′(ω) peaks (10 MHz) of the argyrodites indicated. Asymmetric peaks point to time-scale dependent (heterogeneous) ion dynamics with local jump processes characterised by activation energies as low as 0.115 eV (Li₆PS₅Cl). (c) Temperatures (shown as 1000/T) at which the ρ′(ω) peaks show up. For Li₆PS₅I the peak is clearly shifted toward higher T indicating much slower ion dynamics. The change in E_a,ρ is also shown.

ρ′ is given by a Lorentzian-shaped function containing the electrical relaxation time τ_ρ:⁶⁰

The quantity log₁₀(M′′/ω) when plotted vs. 1/T shows similar features to the diffusion-induced NMR spin–lattice relaxation rate.^15,51,60 The higher the frequency the more the peaks shift towards higher temperatures. Symmetric peaks are obtained for β′ = 2; if β′ < 2 the low-T flank becomes lower than that of the high-T side. Indeed, for Li₆PS₅Br we notice an asymmetric peak that strongly points to correlated motion in disordered ion conductors.^15,61–64 While the high-T flank yields an activation energy E_a,ρ comparable to E_a,DC when bulk processes are probed, short-range motions are seen on the low-T side. The latter motions are characterised by values as low as 0.15 eV, which are also seen for Li₆PS₅Cl and Li₆PS₅Br_0.5I_0.5 (Fig. 10b). E_a,ρ of Li₆PS₅Cl is only slightly higher than E_σ,bulk = 0.305 eV (see above).

We cannot see any strong variation in E_a,ρ when comparing the response of Li₆PS₅Br_0.5I_0.5 with that of Li₆PS₅Br. E_a,ρ values have also been included in Fig. 6; for the sake of clarity, they are separately shown in Fig. 10c. Obviously, a lower pre-factor of the underlying Arrhenius relation is responsible for this shift. The same effect is seen for Li₆PS₅I. Although the lattice is expected to be much softer in the case of Li₆PS₅I as compared to the situation in Li₆PS₅Cl, ρ(ω) tells us that the activation energies E_a,ρ of the two compounds are very similar (0.32 eV vs. 0.35 eV). However, the peak ρ′(ω) of Li₆PS₅I is clearly shifted toward higher T and shows up at 385 K; the origin of this difference has to be looked for in the bulk ion dynamics, g.b. resistances cannot explain it. As compared to the peak of Li₆PS₅Cl the peak of the I compound is symmetric in shape (β ≈ 1.95) and points to an ordered structure.

Obviously, as in the case of E_a,DC the reason for the lower conductivity and, thus, the slower bulk electrical relaxation in Li₆PS₅I (and in Li₆PS₅Br_0.5I_0.5) has been looked for in the prefactor of the underlying Arrhenius relationships. Most likely, the anion site disorder in Li₆PS₅Cl is responsible for this behaviour. Indeed, if we approximate the ρ′(ν) peaks with eqn (3) we can extract pre-factor τ_0,ρ⁻¹ assuming τ_ρ⁻¹ to be depending on temperature according to Arrhenius. τ_0,ρ⁻¹ values are also included in Fig. 6; indeed, they clearly reveal lower pre-factors for Li₆PS₅I by approximately two orders of magnitude. If we assume that the attempt frequencies are similar for Li₆PS₅Cl and Li₆PS₅I a significant change in migration entropy, as mentioned above, might explain the prominent change seen. To check whether this phenomenon is also apparent in SLR NMR, which is in general sensitive to both short-range and long-range ion dynamics, we carried out ⁷Li NMR measurements in both the laboratory and rotating-frame of reference.^64,65 Line shape measurements carried out at temperatures down to the cryogenic region complement our investigation.

3.4 ⁷Li NMR relaxometry to probe the bulk ion dynamics

In Fig. 11 an overview of the ⁷Li NMR spin–lattice relaxation rates 1/T₁ ≡ R₁ and 1/T_1ρ ≡ R_1ρ measured over a wide temperature range is given. To identify the thermally activated regions the rates of Li₆PS₅X, with X = Cl, Br, and I, were plotted using an Arrhenius representation, i.e., log(1/T₁) is plotted vs. 1000/T. The rates have been recorded at a Larmor frequency of ω₀/2π = 116 MHz; for the locking frequency we used a magnetic B₁ field corresponding to ω₁/2π = 20 kHz. At very low temperatures (T < 160 K) the R₁ rates shown in Fig. 11a reveal a non-diffusive background; in this T range longitudinal relaxation is primarily induced by lattice vibrations or coupling of the Li spins (I = 3/2) with paramagnetic impurities.^61,65–67

Fig. 11 (a) Temperature dependence of the diffusion-induced ⁷Li NMR spin–lattice relaxation rates R₁ (116 MHz) of Li₆PS₅X (X: Cl, Br and I). Solid lines refer to BPP-type fits to extract activation energies and pre-factors. (b) Arrhenius plot of the corresponding ⁷Li SLRρ NMR rates recorded in the rotating frame of reference using a nominal locking frequency of ω₁/2π = 20 kHz. Again, the solid lines represent fits according to a modified BPP relaxation model taking into account the asymmetry of the peaks by parameter β. The further the relaxation peak is shifted to lower T, the faster the Li exchange process inducing spin-lock spin–lattice relaxation. Therefore, Li diffusion, as seen by NMR, is increasing in the following order: Li₆PS₅I < Li₆PS₅Br < Li₆PS₅Cl (see also c). (c) R₁ and R_1ρ of Li₆PS₅I; at 160 K Li₆PS₅I reversibly transforms into its low-T phase. At this temperature the rates R₁ show a jump toward lower values; the R_1ρ rates sharply decay with an activation energy of 0.44 eV. The unfilled symbols in (a–c) show rates measured with an NMR probe designed for experiments at cryogenic temperatures; the filled symbols refer to data which were acquired with a standard probe.

At higher T we expect spin–lattice relaxation to be increasingly induced by Li self-diffusion.⁶¹ Indeed, the rates sharply increase with temperature and characteristic diffusion-induced rate peaks show up. We recognise that R₁ passes through well-defined maxima as has been found by some of us in an earlier study on the ion dynamics in Li₆PS₅Br.¹⁵ At the temperature T_max, where R₁ reaches its maximum value, the motional correlation rate 1/τ_c is given by τ_cω₀ ≈ 1. 1/τ_c is identical within a factor of two to the Li jump rate 1/τ.^61,65 The same holds for R_1ρ, for which the maximum condition is ω₁τ ≈ 0.5.^61,65 As ω₀ and ω₁ differ by more than three orders of magnitude, the SLR NMR experiments, when taken together, are able to sense motional processes covering a large time scale.^67,68

We notice that the rate peak of Li₆PS₅Br shows up at T_max = 286 K. The peak is asymmetric and we used a Lorentzian-type (BPP-type)^62,69 spectral density function J(ω₀) ∝ R₁ of the form⁷⁰

to approximate its shape. Eqn (4) holds for 3D diffusion; J(ω₀) is the Fourier transform of the underlying motional correlation function G(t′). If G(t′) is a single exponential, β in J(ω₀) takes a value of 2. For a 3D diffusion process, at high temperatures, i.e., if τ_cω₀ ≪ 1, eqn (4) yields J ∝ τ_c; in the limit τ_cω₀ ≫ 1, that is, on the low-T side of the rate peak, we have J ∝ τ⁻¹ω₀^−β (1 < β ≤ 2). While we expect β = 2 for uncorrelated motion,^61,62,69,71 asymmetric peaks are often found for structurally complex ion conductors^15,71–73 whose dynamic processes are affected by correlation effects because of both disorder and strong Coulomb interactions of the moving ions.⁷¹ For such materials the activation energies for short-range and long-range processes might largely differ.⁶² In general, β can also be related to a distribution width of motional correlation times; in this sense it has a similar meaning to the exponent in the Cole–Davidson spectral density function used to approximate asymmetric rate peaks.^74–76

Here, the analysis of the rate peak of Li₆PS₅Br with eqn (4) yields an activation energy E_a,NMR of 0.213 eV, which is in perfect agreement with earlier findings for this compound.^15,50 As mentioned above, the parameter β describes the deviation from symmetric behaviour (β = 2). β turned out to be β ≈ 1.5,⁶⁵ which gives rise to a much lower activation energy E_a,low on the low-T side of the peak. In this regime, characterised by τ_cω₀ ≪ 1, short-range or localised ion dynamics in double-well potentials, including also highly correlated forward–backward jumps, are sensed by R₁.⁶¹ The activation energy of the low-T flank is given by 0.10 eV, which is in very good agreement with that seen by ρ′(ω). Localised intracage jump processes involving the 48h and 24g sites in Li₆PS₅Br could be mainly responsible for spin–lattice relaxation in this region, as has been precisely calculated (0.11 eV) by De Klerk et al.⁵⁰ The higher activation energy E_a,high(=E_a,NMR) seen in the limit τ_cω₀ ≪ 1 might, however, additionally be affected by 48h–48h jumps. Usually in this regime, in which many jump processes occur during one Larmor precession of the spin, E_a,NMR should be comparable with E_a,DC (0.296 eV) or E_a,ρ (0.250 eV).⁶¹E_a,DC could be influenced by g.b. effects; here, E_a,NMR < E_a,ρ shows that the number of jump events sensed by NMR, which should not necessarily include all types of jumps needed for long-range diffusion, is sufficient to generate a full R₁ peak.

A very similar behaviour is found for Li₆PS₅Cl. The corresponding rate peak is shifted toward higher temperature revealing that faster ion dynamics is present in Li₆PS₅Br. In agreement with conductivity spectroscopy and our ρ′(ω) analysis, for Li₆PS₅Cl higher activation energies (E_a,high = 0.32 eV, E_a,low = 0.17 eV) than those seen for the Br analogue were found. Although E_a,high = 0.32 eV perfectly agrees with E_a,ρ, see Fig. 10, and calculated values,³⁵ it is higher than expected from R_1ρ NMR in the limit τ_cω₁ ≪ 1 (0.248 eV). Furthermore, the shapes of the R₁(1/T) peaks of the two compounds are very similar. Obviously, similar (local) jump processes influence the NMR rates. In Table 1 an overview of the parameters is given obtained from the analysis with individual BBP-type spectral densities for each peak.

Table 1 Results of the individual BPP-fits used to analyse the ⁷Li NMR R₁ and R_1ρ peaks of the three argyrodite samples Li₆PS₅X (X = Cl, Br, I). For comparison, the activation energies obtained from the low-T flank, E_a,low, are also included

	Li6PS5Cl^a	Li6PS5Br^a	Li6PS5I^b
a The best R1ρ fit is obtained if we replace ω1/2π = 20 kHz by a slightly higher effective locking frequency ωeff that also takes into account local magnetic fields. For Li6PS5Cl several runs of the global fit procedure yields ωeff = 1.68 × ω1; for Li6PS5Br ωeff is given by ωeff = 1.41 × ω1. b Li6PS5I undergoes a phase transition at 160 K; thus, no βρ can be observed. The symmetric rate peak (β ≈ 2) agrees with the ordered anion sublattice as suggested by XRPD, see above.
β	1.53(2)	1.48(2)	1.92(1)
E a	0.320(1) eV	0.213(1) eV	0.198(1) eV
E a,low	0.170(4) eV	0.102(5) eV	0.182(5) eV
τ 0	1.3(5) × 10^−14 s	2.3(5) × 10^−13 s	8.2(5) × 10^−13 s
β ρ	1.54	1.41	—
E a,ρ	0.248(9) eV	0.201(9) eV	0.137(3) eV
E a,low	0.134(1) eV	0.083(1) eV	—
τ 0,ρ	9.5(5) × 10^−13 s	3.3(1) × 10^−12 s	—

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Exactly the same trend is seen when we look at the R_1ρ(1/T) peaks of the two compounds, which are shown in Fig. 11b. These have been, independently of the R₁ experiments, also parameterised with Lorentzian-shaped spectral density functions. The activation energies of the peak of Li₆PS₅Br (0.201 eV, 0.083 eV) match with those reported earlier¹⁵ also including calculated values.⁵⁰ They also agree with those from the R₁ peak. Here, R_1ρ(1/T) of Li₆PS₅Br turned out to be much smaller that that reported by Yu et al.;⁷⁷ most likely, the difference has to be looked for in the preparation conditions such as milling steps and annealing procedures.

In Fig. 12 so-called joint fits⁶⁵ of the two types of NMR experiments, R₁ and R_1ρ, are shown. For the joint fits the parameters E_a,NMR and τ₀⁻¹, which is the pre-factor of the Arrhenius equation of τ_c⁻¹, have been linked to each other. The best fit is obtained if we use two independent values for β and if we replace ω₁ by effective frequencies ω_1,eff, also taking into account local magnetic fields.⁷³ To estimate the latter, we did not fix ω₁ and looked at how the quality of the fit changes if it is a free parameter. Only a slight change of ω₁ was necessary giving rise to ω_eff ≈ 1.5 × ω₁. A successful joint fit shows that the same overall diffusion processes govern the peaks R₁ and R_1ρ. The results of our joint fits are shown in Table 2. The activation energies E_a,NMR (=E_a,high) obtained fulfill the relationship

E_a,low = (β_(ρ) − 1)E_a,high. (5)

β_(ρ) < 2 points to correlated Li⁺ motion and indicates a sub-quadratic dependence of the diffusion-induced SLR NMR rates in the low-T limit τ_cω₀₍₁₎ ≫ 1. Our results are in good agreement with those of Epp et al. as well as Rao et al.^15,35 We clearly see that correlation effects such as structural disorder and Coulomb interactions⁷¹ have a larger effect on the cation dynamics in Li₆PS₅Br than in Li₆PS₅Cl.

Fig. 12 Arrhenius plot of the ⁷Li NMR relaxation rates R₁ and R_1ρ of (a) Li₆PS₅Cl and (b) Li₆PS₅Br measured in the laboratory frame of reference (116 MHz) and in the rotating frame of reference (20 kHz, nominal locking frequency). The solid lines represent the global fit based on a modified BPP model that takes into account peak asymmetries by the parameter β_(ρ). The fastest Li diffusion is found for Li₆PS₅Br with the peak maxima showing up at 286 K and 168 K, respectively. The second R_1ρ relaxation process, most likely also influenced by spin–spin relaxation to a certain degree, is indicated, too. As mentioned before (see Fig. 11), the open symbols represent rates that were measured with an NMR probe designed for experiments at cryogenic temperatures; the filled symbols show rates which were acquired with the standard probe.

Table 2 Results of the overall fit used to jointly parameterise the ⁷Li NMR R₁ and R_1ρ peaks of Li₆PS₅Br and Li₆PS₅Cl. E_a agrees well with the results if we analyse the peaks individually. For comparison, results for Li₆PS₅I taken from Table 1 are also included

	E a,NMR	C	β (ρ)	τ 0
a The best global fits are obtained if we replace ω1 by the effective locking frequencies ωeff = 1.68 × ω1 (Li6PS5Cl) and ωeff = 1.41 × ω1 (Li6PS5Br). The coupling constant C, which is the amplitude of the rate peak, J(ω0) = Cτc/(1 + (ωτc)^β), turns out to be on the order of 10^9 to 10^10 s^−2.
Li6PS5Cl
R 1	0.273(5) eV	3.5(5) × 10^9 s^−2	1.60(1)	4.9(4) × 10^−14 s
R 1ρ		3.1(5) × 10^10 s^−2	1.47(1)
Li6PS5Br
R 1	0.218(1) eV	5.6(7) × 10^9 s^−2	1.48(1)	2.0(2) × 10^−13 s
R 1ρ		3.3(9) × 10^10 s^−2	1.47(1)
Li6PS5I^a
R 1	0.198(1) eV	1.1(4) × 10^10 s^−2	1.92(1)	8.2(9) ×10^−13 s
R 1ρ	0.137(3) eV	—	—	—

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Considering the overall constants C obtained from the joint fits, they reveal that both dipolar and electric quadrupolar interactions govern the diffusion-induced SLR NMR rates. Estimating C_dipolar with the help of the rigid lattice line width for Li₆PS₅Br and Li₆PS₅Cl would underestimate C to be on the order of 10¹⁰ s⁻².⁴⁶ As we will discuss below, for Li₆PS₅Br quadrupolar interactions have been identified to greatly influence longitudinal relaxation.¹⁵

Interestingly, a very close look at the R_1ρ magnetization transients revealed that they contain a minor magnetization component that leads to activation energies of 0.484 eV (Li₆PS₅Cl) and 0.298 eV (Li₆PS₅Br); the corresponding R_1ρ rates are included in Fig. 12. We cannot exclude that these rates are to a certain degree also influenced by spin–spin relaxation. Obviously, the rates sense long-range Li jump processes characterised by higher activation energies. The intercage jumps might contribute to this type of magnetic relaxation, too.

For X = Cl and Br, if we consider activation energies, T_max, β and β_ρ, the ⁷Li SLR NMR data reveal the same trend for the ion dynamics as seen by the conductivity and ρ′(ω). Li₆PS₅Br shows fast ion dynamics and the lowest activation energies. For Li₆PS₅Cl we witness slightly higher activation energies; simultaneously, the peaks shift toward higher temperature. For Li₆PS₅I, on the other hand, significant differences show up (see the next section).

The NMR rate peaks of Li₆PS₅I, remembering its low σ_DC conductivity of 10⁻⁶ S cm⁻¹ at ambient temperature, are expected to show up at temperatures much higher than 320 K. We would expect much higher activation energies, at least similar to or higher than those found for Li₆PS₅Cl. Surprisingly, the R₁ rate peak of Li₆PS₅I shows up at a temperature highly comparable to that of Li₆PS₅Cl viz. at T_max = 329 K, see Fig. 11a. In stark contrast to our expectation, NMR does not point to slower Li⁺ diffusion. Hence, in Li₆PS₅I the same fast (local) jump processes are present as in their parent compounds.^35,40,50 Here, we anticipate that the Li ions in Li₆PS₅I, with its ordered anion sublattice, have access to the same rapid intracage jump processes. According to NMR the barriers have to be characterised by an activation energy of 0.2 eV. Obviously, only intracage jump processes in Li₆PS₅I are sufficient to produce a full R₁(1/T) peak, see the discussion above.

As expected for a material with an ordered anion lattice such as Li₆PS₅I and without severe influences of motional correlation effects, an almost symmetric rate peak R₁(1/T) (β = 1.92(≈β′)) is obtained for Li₆PS₅I. The same shape was seen in ρ′(ω). Interestingly, an even lower activation energy determines R_1ρ(1/T), see Fig. 11c: in the limit τ_cω₁ ≪ 1 the rates follow an Arrhenius line whose activation energy E_a,high is given by only 0.137 eV. This value is remarkably similar to a cage-like local pathway studied by Rao and Adams (0.15 eV) who used a bond valence approach to investigate the Li ion dynamics in Li₆PS₅I. Our experimental value also coincides with that of Pecher et al.⁴⁰ (0.14 eV) who studied the local ion dynamics by molecular dynamics simulations.

The amplitude (R₁)_max of the R₁(1/T) rate of Li₆PS₅I turned out to be higher that that of the peaks belonging to Li₆PS₅Cl and Li₆PS₅Br. This could be because of stronger heteronuclear Li–X interactions and the fact that Li ions on the distorted 24g sites give rise to stronger dipolar and electric quadrupolar couplings constants. These are lowest for the Li₆PS₅Cl compound. In general, (R₁)_max is proportional to the square of the magnetic dipolar or electric quadrupolar coupling constant.^75,78,79 If we use an electric quadrupole constant of at least 50 kHz, as has been estimated for Li₇PS₆,⁴⁶ we obtain (R₁)_max ≈ 2 s⁻¹ yielding log₁₀((R₁)_max/s) ≈ 0.3, showing that for Li₆PS₅Cl both dipolar and electric quadrupolar interactions play a role in determining the relaxation mechanism. For Li₆PS₅Br we showed via comparative ⁷Li and ⁶Li SLR NMR measurements that quadrupole fluctuations greatly affect the overall spin–lattice relaxation.¹⁵

Coming back to the temperature dependence of the SLR NMR rates of Li₆PS₅I, we see that at higher temperatures than ambient the rates R_1ρ are increasingly governed by R₁, which produces an apparent maximum at ca. 330 K. Unfortunately, the peak maximum of R_1ρ(1/T) cannot be detected as Li₆PS₅I undergoes a phase transition at 160 K;^31,39,40,49 see the abrupt change in R_1ρ in Fig. 11c. The ion dynamics in the low-T (LT) phase is characterised by a much higher activation energy of 0.44 eV. Below 160 K the rates R₁ are no longer induced by diffusive Li jump processes.

To explain the evident differences between the results from conductivity measurements and nuclear spin relaxation in Li₆PS₅I we anticipate that the fast diffusion Li processes in the I compound are not interconnected to give rise to long-range, through-going ion transport. As has been pointed out in detail by Wagemaker and co-workers, and first mentioned by Rao and Adams,³⁵ the jump probability between two rings in Li₆PS₅I is very low.⁵⁰ Intercage jump processes with higher activation energies lead to poor ion transport in Li₆PS₅I. On a shorter length scale, however, the same structural motifs in Li₆PS₅X cause very similar NMR rate peaks. Anion disorder, taking advantage of anions with radii that do not differ much to that of S²⁻, switches on the intercage exchange processes, making Li₆PS₅Br a fast ion conductor. This design principle causes the overall activation energy to increase in the following order: Li₆PS₅Br < Li₆PS₅Cl < Li₆PS₅I.

To verify whether some intragrain translational processes needed for long-range diffusion in Li₆PS₅I are indeed missing or governed by higher activation energies we carefully looked at the R_1ρ transients and recorded ⁷Li NMR line shapes over a broad temperature range (Fig. 13). Interestingly, the above-mentioned slow magnetization component seen for Li₆PS₅Cl and Li₆PS₅Br (Fig. 12) is missing for the I compound. Furthermore, the intercage jumps are expected to contribute to the averaging of homonuclear (Li–Li) dipole–dipole interactions. If missing at low temperatures, for Li₆PS₅I the change in line width Δν should occur in two steps. Full averaging is expected at temperatures for which the intercage exchange rate reaches sufficiently high values on the order of some kHz. Indeed, whereas for Li₆PS₅Br and Li₆PS₅Cl the ⁷Li NMR line width continuously decreases with temperature revealing typical narrowing curves (Fig. 13a), that of Li₆PS₅I is different.

Fig. 13 (a) Temperature dependence of the ⁷Li NMR line widths (FWHM = full width at half maximum) of the aryrodites Li₆PS₅X (X: Cl, Br and I). Some of the data points referring to Li₆PS₅Br and measured at temperatures lower than 80 K were taken from Epp et al.;¹⁵ they perfectly complement the results of the sample studied here. Line widths Δν_r(T) of Li₆PS₅Br start to narrow at temperatures as low as 86 K indicating ultrafast hopping processes that average homonuclear dipole–dipole interactions being responsible for the rigid lattice line width Δν_r ≈ 6.1 kHz. Li diffusion in the compound with Cl is slightly slower as the onset of line narrowing is shifted toward higher T. The line of Δν_r(T) of Li₆PS₅I narrows in steps of two. At T < 160 kHz it indicates dipole averaging in the low-T phase. Because inter-ring jump processes are absent, the line does not reach its extreme value at a temperature comparable to that of Li₆PS₅Br and Li₆PS₅Cl. Higher temperatures are necessary to obtain a fully narrowed ⁷NMR line. Lines represent fits according to the models of Hendrickson and Bray (dashed lines) and Abragam (solid lines). (b) ⁷NMR line shapes of Li₆PS₅Br recorded at the temperatures indicated; the line transforms from a Gaussian shape at low T to a Lorentzian one at elevated temperatures.

For Li₆PS₅Br and Li₆PS₅Cl the extreme narrowing regime is already reached at 250 K (ν_∞ ≈ 300 Hz); the final line width is caused by external field inhomogeneities. In contrast to the situation observed for Li₆PS₅Br and Li₆PS₅Cl, at 250 K the ⁷Li NMR line of Li₆PS₅I is still relatively broad and shows a width of ν = 2.2 kHz. At 350 K it finally reaches ν_∞. Thus, for Li₆PS₅I full averaging of the NMR line width is shifted by ΔT = 100 K toward higher T. Most likely, the latter step corresponds to intercage ion dynamics which are governed by a larger activation barrier. Hence, it is clear from NMR motional narrowing data that in Li₆PS₅I a much slower process kicks in only at higher T. The fact that Δν of Li₆PS₅I and Li₆PS₅Cl is very similar, at least at temperatures just above the phase transition, shows that mainly fast intracage jump processes are responsible for the narrowing seen in this T range. Differences show up, however, if we consider the rotating-frame spin–lattice relaxation rates, see above. The activation energies of the two samples differ by ca. 0.11 eV. Interestingly, the R_1ρ(1/T) peak of Li₆PS₅I is expected at a similar temperature to that of Li₆PS₅Br (0.20 eV), again pointing to very fast intracage jump processes in the iodide compound. Most likely, so-called doublet-jump processes⁵⁰ have to be considered to explain this feature, see below.

Unfortunately, we cannot probe the rigid-lattice line width ν₀ of the Li₆PS₅I compound because of the phase transition that occurs at 160 K.^31,39,40,49 For both Li₆PS₅Br and Li₆PS₅Cl the value of ν₀ is very similar; ν₀ turned out to be ca. 6.1 kHz. Therefore, the magnetic dipolar Li–Li interactions, which are determined by the Li–Li distances, are almost the same in the two compounds. The inflexion point of the motional narrowing curve of Li₆PS₅Br is located at T = 150 K (Fig. 13a). R_1ρ(1/T) tells us that at this temperature the motional correlation rate τ_c⁻¹ (≈2ω₁) should be on the order of 2.51 × 10⁵ s⁻¹. This value clearly exceeds ν₀ and causes full averaging of the spin–spin interactions in this T range. The changes of the corresponding ⁷Li NMR lines are shown in Fig. 13b. Li₆PS₅Br shows one of the lowest onset temperatures seen by NMR motional line narrowing; already at 86 K the shape of the NMR line starts to change as a result of rapid Li⁺ self-diffusion.¹⁵ This observation excellently agrees with results from density functional theory molecular dynamics simulations.⁵⁰

Extracting quantitative information from NMR motional line narrowing is always fraught with difficulties.⁸⁰ Abragam⁸¹ and Hendrickson and Bray^82,83 introduced models to use the change in line widths to deduce activation energies for the hopping process behind. There is still debate over whether these models are accurate enough for this purpose. Hence, we should not overinterpret the results but regard them as estimates. The dashed lines in Fig. 13 refer to the following relationships⁸²

Δν(T) is the line width of the central transition at a given temperature and B is a parameter that is related to the number of thermally activated ions.⁸² The final line width ν_∞ is determined by the inhomogeneities of the external magnetic field B₀. Another possibility to deduce activation energies is given by the formalism of Abragam⁸¹ (see the solid line in Fig. 13):α is a fitting parameter, here chosen to be 1. τ_∞ is the pre-factor of the underlying Arrhenius law of the correlation time τ_c. With the Abragam fit we obtained activation energies E_a,A of 0.06 eV (Li₆PS₅Br), 0.11 eV (Li₆PS₅Cl) and 0.23 eV (Li₆PS₅I), respectively. Typically, the formalism of Hendrickson and Bray yields somewhat higher activation energies; here, we found E_a,HB = 1.6 × E_a,A, see Table 3. Interestingly, although line narrowing is expected to be sensitive to long-range ion dynamics which can fully average all dipolar interactions, the activation energies E_a,A and E_a,HB are remarkably similar to those governing the low-T flank of the R₁(1/T) peaks, see also Table 3.

Table 3 Activation energies obtained when analysing the motional narrowing curves shown in Fig. 13. We used the Abragam (A) and Hendrickson–Bray (HB) formalism to extract activation energies; the error is in the last digit. For comparison, activation energies describing the slopes of the R₁(1/T) peaks in the limit τ_cω₀ ≫ 1 are also shown

	E a,A (eV)	E a,HB (eV)	E a (low-T flank) (eV)
Li6PS5Cl	0.11(1)	0.18(2)	0.17(4)
Li6PS5Br	0.06(1)	0.09(1)	0.10(4)
Li6PS5I	0.23(1)	0.38(2)	0.18(5)

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Finally, we will quantitatively compare jump rates τ⁻¹ from NMR with electrical relaxation rates from impedance and conductivity spectroscopy. Using the Nernst–Einstein equation⁸⁴ we can convert σ′ into diffusion coefficients and, thus, jump rates. The easiest way, being also independent of any relaxation model, is to determine jump rates τ⁻¹ at the temperatures where the NMR relaxation rate peaks show up. With the conditions τ_cω_1(ρ) = 1(0.5), valid at T_max, and the jump distances [small script l] obtained from XRD we estimated diffusion coefficients according to the Einstein–Smoluchowski relation⁸⁴ for 3D diffusion

For [small script l] we used the jump distance of the 48h–48h jump process, which is assumed to be the slowest one. The results are listed in Table 4. At T_max the jump rate turned out to be on the order of 1.36 × 10⁻⁹ s⁻¹. As an example, the mean residence time of the Li ions in Li₆PS₅Br at 286 K is on the order of 1 ns only. Such a short residence time, which is already reached below room temperature, corresponds to ionic conductivities on the order of 10⁻³ to 10⁻² S cm⁻¹.

Table 4 Results of the diffusion coefficient D. τ was calculated from the R₁(1/T) rate peak measurements performed at 116 MHz

	[small script l]/Å	T max(R1)/K	D NMR/cm^2 s^−1	n Li/m^−3
For [small script l] we used the 48h–48h distance, which is the longest jump distance in Li6PS5Br, when we consider cation hopping processes near the regularly occupied Li sites.
Li6PS5Cl	2.34	316	6.7 × 10^−8	6.3 × 10^27
Li6PS5Br	2.41	286	7.1 × 10^−8	6.0 × 10^27
Li6PS5I	2.66	329	8.6 × 10^−8	5.8 × 10^27

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Using structural data from our Rietveld analysis we also estimated the charge carrier density n_Li in Li₆PS₅Br. n_Li, which is on the order of 10²⁷ m⁻³, is needed to convert D_NMR into conductivities expected at or near T_max. Disregarding any correlation effects for our estimation of σ_NMR, the following relation holds¹¹

where e denotes the elementary charge. Here, a residence time τ of 1 ns results in conductivities of the order of 2 mS cm⁻¹. In Table 5 the estimated results from NMR are compared with those measured by conductivity spectroscopy. Assuming no correlation effects or other influences for this estimation means that H_R/f in D_NMR = (H_R/f)D_σ is approximated by H_R/f = 1; while H_R denotes the Haven ratio connecting the tracer diffusion coefficient D_tr with D_σ, the correlation factor f links D_NMR with D_tr.¹¹

Table 5 Estimated conductivity values (σ_NMR) from NMR data. For comparison, the measured values σ_DC are also listed

	σ NMR/mS cm^−1	σ DC/mS cm^−1
Li6PS5Cl	2.46 (316 K)	9.00 (313 K)
Li6PS5Br	2.78 (286 K)	2.20 (293 K)
Li6PS5I	2.80 (329 K)	8.60 × 10^−3 (333 K)

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While good agreement is seen for σ_NMR and σ_DC of Li₆PS₅Br, we have to notice a somewhat larger difference for the Cl compound. In general, conductivities indirectly deduced from NMR should be regarded as estimates because relaxation NMR and conductivity spectroscopy sense different kinds of motional correlation functions. Agreement over a large temperature is only obtained if both techniques are sensitive to the same diffusion process and the same correlation function.⁵¹

As expected, for Li₆PS₅I the values listed in Table 5 differ by three orders of magnitude. Whereas σ_DC is mainly governed by successful jump processes resulting in long-range ion transport, σ_NMR reflects short-range ion dynamics. The difference can easily be explained by the fact that in Li₆PS₅I the intercage jump processes are less probable; these processes hinder the ions in long-range diffusion. Note that the comparison in Table 5 only refers to T ≈ T_max at which the NMR peaks show up. Comparing dynamic parameters over the full temperature range will show deviations between nuclear spin relaxation and conductivity spectroscopy as the techniques deliver different activation energies in the present case. Such a comparison is presented in Fig. 14. Squares represent jump rates obtained at the NMR rate peaks; at T → 0 we have indicated the pre-factors of the Arrhenius lines.

Fig. 14 Arrhenius plot of the jump rates deduced from the ⁷Li NMR rate peaks shown in Fig. 11c, 12a and (b). (a) Squares represent either pre-factors or jump rates at the maxima of the peaks. Solid lines represent the Arrhenius lines according to the joint fits. Dashed lines indicate the position of τ_σ⁻¹, i.e., estimated from σ_DC. (b) The same data points as in (a) but with τ_DC⁻¹ included as data points. The lines referring to Li₆PS₅Cl and Li₆PS₅Br have been shifted by a factor of +4 (+2) (on the log scale) to illustrate the kink seen in τ_DC⁻¹(1/T) for Li₆PS₅Cl. It is absent for Li₆PS₅Br. Electrical relaxation rates τ_ρ⁻¹ from our ρ′(1/T) analyses have been included as well. The vertical arrow drawn with a dashed line indicates the large difference in jump rates seen for Li₆PS₅I. The arrows near the ordinate illustrate the change in pre-factors. See the text for further explanation.

In Fig. 14aτ_NMR⁻¹ as deduced from the joint fits of Li₆PS₅Cl and Li₆PS₅Br are shown. The line belonging to Li₆PS₅I refers to the R₁ rates only. First of all, we notice a correlation between E_a,NMR and the pre-factor τ₀⁻¹ (see also Table 2). The decrease in E_a,NMR, which changes from 0.273 eV to 0.218 eV and further to 0.198 eV (R₁ of Li₆PS₅I), is accompanied by a decrease in τ₀⁻¹. The attempt frequency τ₀⁻¹ decreases by a factor of 80 when going from Li₆PS₅Cl to Li₆PS₅I. This small variation either reflects a decrease in phonon frequencies or a change in migration entropy ΔS of the ions as τ₀⁻¹ = ν_p,0exp(ΔS/k_B). Phonon frequencies, to which ν_p,0 is proportional, usually take values on the order of 10¹² to 10¹⁵ s⁻¹. As discussed above, for Li₆PS₅I a decrease in migration entropy is reasonable because of the ordered anion sublattice. The decrease in E_a,NMR reflects the increase in anion polarizability (I > Br > Cl), which should, apart from effects due to disorder, also lead to lower hopping barriers. This trend for E_a,NMR is at least valid for the elementary Li⁺ jumps seen by NMR. If we include τ_σ⁻¹ values obtained by transforming σ_DC into hopping rates after eqn (9), we notice that good agreement is only seen at temperatures around T_max, see Fig. 14a and b. Clearly, the fact that E_a,NMR ≠ E_a,DC leads to deviations at temperatures T ≶ T_max. For comparison, in Fig. 14b hopping rates from ρ′(T) have also been included.

The pre-factors of the Arrhenius lines referring to τ_σ⁻¹ estimated from σ_DC also vary with X. While we cannot see any large difference between τ_0,σ⁻¹ from Li₆PS₅I and Li₆PS₅Br as mentioned above, we see that τ_0,σ⁻¹ of the Cl-compound is much higher than expected. This increase is caused by a change in slope of σ_DC at a temperature slightly below 400 K; it is also seen for Li₆PS₅I. Below 400 K the ionic conductivity σ_DC (as well as τ_σ⁻¹) follows an Arrhenius law with an activation energy very similar to that of Li₆PS₅Br (0.273 eV). As σ_DC of Li₆PS₅Br and Li₆PS₅Cl almost coincide in this T range, also the corresponding pre-factors τ_0,σ⁻¹ would be the same. Here, the fact that τ_0,σ⁻¹ of Li₆PS₅Cl is larger than that of Li₆PS₅Br originates the kink in the Arrhenius line causing σ_DCT(1/T) to be characterised by an activation energy (0.396 eV) much larger than that in the low-T range. Obviously, another Li⁺ diffusion process, being characterised by an activation energy larger than 0.3 eV, contributes to τ_σ⁻¹ of Li₆PS₅Cl.

Such an additional process, which we assume could stem from an ordered Li sublattice in Li₆PS₅Cl, has been seen by R_1ρ, see Fig. 12. The R_1ρ magnetization transients of Li₆PS₅Cl are characterised by a second decay process that is thermally activated by 0.48 eV. The same feature is also seen for Li₆PS₅Br with both a disordered anion structure and Li⁺ sublattice. The corresponding activation energy of 0.30 eV (see Fig. 12) is, however, very similar to that of σ_DCT(1/T). This similarity thus gives rise to no increase in σ_DCT above T ≈ 400 K.

The pre-factors seen by conductivity spectroscopy, especially those of Li₆PS₅Cl, clearly represent mean values. In materials with several diffusion processes taking place in parallel, each process needs to be characterised by its individual pre-factor. In the case of ⁷Li NMR, which is primarily sensitive to the elementary jump processes between 24g and 48h, we see a slight increase in the pre-factor when going from X = I to X = Cl (see above). The change in τ₀⁻¹ is, however, small as compared to the drastic decrease in σ_DCT(1/T) seen for Li₆PS₅I. Hence, the increase in overall migration enthalpy for Li₆PS₅I, for which the intercage diffusion process is characterised by a high hopping barrier, turned out to be mainly responsible for its poor ionic conductivity. On shorter length scales, Li ion hopping in Li₆PS₅I, with its ordered anion sublattice, is as fast as in Li₆PS₅Br. This result is in perfect agreement with the pioneering calculations by Rao and Adams as well as the in-depth study presented by De Klerk et al.⁵⁰ They already pointed out, as mentioned above, that for Li₆PS₅I the intercage jump processes are interrupted and fast localized motions with very low activation energies are present. By ⁷Li NMR relaxometry their observation is now fully supported. The calculations by Pecher et al. point in the same direction.⁴⁰ As an example, De Klerk et al. report an activation energy as low as 0.05 eV for the doublet jump process in Li₆PS₅I.⁵⁰

Besides these peculiarities for Li₆PS₅Cl and Li₆PS₅I, the complex ion dynamics in Li-bearing argyrodites with anion disorder is reflected in the distribution of activation energies. Table 6 summarises the activation energies obtained by the different methods applied. Depending on the length-scale and time-scale to which the specific method is sensitive quite different activation energies can be probed. Such a result is typical for ionic conductors whose charge carriers are exposed to an irregularly shaped energy landscape that gives rise to highly correlated forward and backward motions or fast (local) dynamics on the angstrom length scale.

Table 6 Activation energies as deduced from the various methods applied here to investigate the ion dynamics in Li₆PS₅X by conductivity spectroscopy and NMR relaxometry

	Li6PS5Cl (eV)	Li6PS5Br (eV)	Li6PS5I (eV)
a The values referring either to the high-T or to the low-T flank were obtained by a line fit. b Results from the global fit analysis are obtained by linking Ea and τ0^−1 of the individual relaxation rate peaks. Values in brackets refer to the low-T flanks of the global fit.
σ DC	0.396(3)	0.296(2)	0.470(5)
ρ′, high-T flank^a	0.321(3)	0.250(3)	0.347(7)
ρ′, low-T flank^a	0.153(2)	0.115(9)	0.305(2)
M′′	—	—	0.349(5)
R 1, high-T flank^a	0.320(1)	0.213(1)	0.198(1)
R 1, low-T flank^a	0.170(4)	0.102(4)	0.182(5)
R 1ρ, high-T flank^a	0.248(9)	0.201(9)	0.137(3)
R 1ρ, low-T flank^a	0.134(1)	0.083(1)	—
R 1 (global fit)^b	0.273(5)	0.218(1)	—
	(0.165(8))	(0.105(4))	—
R 1ρ (global fit)^b	0.273(5)	0.218(1)	—
	(0.129(7))	(0.081(4))	—

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4 Summary and conclusion

We studied the Li ion dynamics in argyrodite-type Li₆PS₅X (X: Cl, Br, I) by a wide range of complementary techniques with a special focus on conductivity spectroscopy and nuclear magnetic relaxation. The latter technique is able to probe short-range as well as long-range bulk ion dynamics. The analysis of electric modulus data also helped us to shed light on the bulk ion dynamics. In all compounds, even in those where two different halogen atoms were used to replace a sulfur anion per formula unit, complex ion dynamics is found. Heterogenous Li ion transport is traced back to the fact of an irregularly shaped potential landscape that the ions are exposed to during diffusion. The vacancy-rich Li sublattice offers a range of hopping barriers between the Li sites involved in the overall ionic transport. Depending on the method applied to study the hopping processes, Li ion translational motions on quite different time scales and lengths scales are probed. This circumstance results in activation energies ranging from 0.08 eV to values as high as 0.48 eV.

Anion disorder in Li₆PS₅Br and Li₆PS₅Cl, as verified by X-ray powder diffraction measurements, boosts the ion dynamics as compared to the parent compound Li₇PS₆. The halogen anions are clearly distributed over the crystallographic sites 4a and 4d. Li₆PS₅Br reveals the lowest activation energies and the bulk ion conductivities follow a single Arrhenius line. Deviations from this behaviour are seen for Li₆PS₅Cl and Li₆PS₅I. For Li₆PS₅Cl an additional dynamic process shows up at higher T, which is also seen in ⁷Li spin-lock NMR spectroscopy. This process leads to a relatively high mean activation energy in σ_DC characterising the overall ion transport.

Most importantly, also for the poor ionic conductor Li₆PS₅I our NMR relaxation measurements reveal very fast Li ion dynamics on a local to medium-range length scale. Obviously, the ions have access to the same, rapid exchange processes as in Li₆PS₅Br but long-range transport is switched off. In the spirit of De Klerk and co-workers we think that the ordered anion sublattice, combined with the larger lattice constant, is responsible for this peculiarity. It leads to a heterogeneous potential landscape with low and high hopping barriers. High activation barriers seem to characterize the intercage jumps; these jump processes are, however, necessary to transport the ions over long distances.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Financial support by the Deutsche Forschungsgemeinschaft (DFG Research Unit 1277, grant no. WI3600/2-1(4-1) (FOR1277)) as well as by the Austrian Federal Ministry of Science, Research and Economy, and the Austrian National Foundation for Research, Technology and Development (Christian Doppler Laboratory of Lithium Batteries: Ageing Effects, Technology and New Materials) is greatly appreciated. Furthermore, we thank the FFG in the frame of the K project safe battery.

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Footnote

† Electronic supplementary information (ESI) available: Rietveld refinements and structural data, further NMR data. See DOI: 10.1039/c9cp00664h

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