Aluminium ion doping mechanism of lithium thiophosphate based solid electrolytes revealed with solid-state NMR

Abstract

We investigate the impact of Al incorporation on the structure and dynamics of Al-doped lithium thiophosphates (Li_3−3xAl_xPS₄) based on β-Li₃PS₄. ²⁷Al and ⁶Li magic-angle spinning NMR spectra confirm that Al³⁺ ions occupy octahedral sites in the structure. Quantitative analyses of ²⁷Al NMR spectra show that the maximum Al incorporation is x = 0.06 in Li_3−3xAl_xPS₄. The ionic conductivity of β-Li₃PS₄ is enhanced by over a factor 3 due to Al incorporation. Further increase of the Al doping level leads to the formation of a more complicated material consisting of multiple crystalline and distorted phases as indicated by ³¹P NMR spectra and powder X-ray diffraction. Consequently, novel Li ion diffusion pathways develop leading to a very high ionic conductivity at room temperature. NMR relaxometry shows that the activation barrier for long-range Li ion diffusion in β-Li₃PS₄ hardly changes upon Al incorporation, but the onset temperature for motional narrowing comes down significantly due to Al doping. The activation barrier in the subsequently formed multiphase material decreases significantly, however, indicating a different more efficient Li ion conduction pathway.

---

1. Introduction

All-solid-state lithium batteries (ASSLBs) are recognized as promising next generation energy storage systems as they have the potential to provide higher energy density and assured safety.^1,2 To realize this, a lot of work has been done to develop high ion-conductive solid electrolytes. A variety of electrolytes have been investigated, including oxides, sulphides, halides, etc.^3,4 Among these, thio-LISICON (LIthium SuperIonic CONductor) families with a general formula of (M = Ge, Si, Sn, etc. M′ = P, Ga, Al, Zn, etc.) are of great interest due to their high ionic conductivity, reaching 10⁻⁴–10⁻² S cm⁻¹ at ambient temperature which is comparable, or even superior, to the performance of liquid electrolytes.^5,6

Li ion diffusion within solids is usually governed by the presence of Li defects, i.e. vacancies and/or interstitials. The number of mobile species and vacancies determines the ionic conductivity.⁷ Aliovalent substitution is an easy but effective way to modify certain materials. Ions of the host structure are replaced by ions of a different valence, introducing additional vacancies or interstitials elsewhere in the structure to preserve electroneutrality.⁸ According to this rule, thio-LISICON families are derived from Li₄GeS₄via various cation substitutions at the Ge⁴⁺ site.^6,9,10 For instance, partial replacement of Ge⁴⁺ by P⁵⁺ (Ge⁴⁺ + Li⁺ → P⁵⁺) creates vacancies in the Li sub-lattice and generates a class of solid ion conductors with extremely high ionic conductivity: Li_4−xGe_1−xP_xS₄ (LGPS), including Li_3.25Ge_0.75P_0.25S₄ (2.2 mS cm⁻¹ at 25 °C) and Li_3.33Ge_0.33P_0.67S₄ (commonly known as Li₁₀GeP₂S₁₂, 12 mS cm⁻¹ at 25 °C).^6,11 Unfortunately, the high cost and chemical instability of germanium renders the application of LGPS impractical.^12,13

Al doping has been widely used in oxide electrolytes to stabilize the structure and enhance the Li ionic conductivity since aluminium is abundant and stable.^14–18Fig. 1 displays the crystal structure of β-Li₃PS₄ and several building blocks such as PS₄³⁻, P₂S₆⁴⁻, and P₂S₇⁴⁻ which are commonly observed in the lithium thiophosphate family.^19,20 β-Li₃PS₄ has an orthorhombic structure characterized by space group Pnma (No. 62). It consists of isolated PS₄³⁻ tetrahedra. Li ions are distributed over 3 types of Wyckoff sites, namely, tetrahedral Li1 (8d), octahedral Li2 (4b) and tetrahedral Li3 (4c) with site occupancies of 1.0, 0.66 and 0.34, respectively. Li ion diffusion in β-Li₃PS₄ is highly anisotropic along the b axis via the 4b–4c sites as these sites are not fully occupied, i.e. contain many vacancies facilitating Li hopping.^21–23

Fig. 1 Crystal structure of β-Li₃PS₄ and some common P–S units in Li₂S–P₂S₅ binary system. The crystal structure is created by VESTA.²⁶ The Li, P, and S atoms are represented by grey, orange, and yellow spheres, respectively. The half orange P atoms indicate that only the diagonal P sites can be occupied simultaneously.

Previous studies of amorphous Li–Al–P–S systems showed that the addition of Al can induce a structural transformation.²⁴ To the best of our knowledge, the effects of incorporating Al on the Li ion dynamics and structure of crystalline β-Li₃PS₄ has not been explored in the literature. In this paper, we focus on understanding the Al ion doping mechanism and Li ion dynamics using solid-state NMR spectroscopy focussing in particular on the Al incorporation in β-Li₃PS₄ and the structural development of the samples as a function of Al addition. We perform ^6/7Li, ²⁷Al and ³¹P NMR to explore Al incorporation in β-Li₃PS₄ to establish the relationship between Li ion dynamics and the structure of the Al-doped materials. It is found that Al dopants occupy octahedral Li2 (4b) sites, introducing vacancies at the Li1 (8d) sites which are fully occupied in pristine β-Li₃PS₄. Therefore, Li ion jumps between bc planes are facilitated. Only part of the Al from the Al₂S₃ precursor enters the β-Li₃PS₄ system. We also show that the addition of Al induces further structural evolution, forming multiphasic material consisting of both crystalline and amorphous components that substantially facilitate Li ion transport. The electrochemical characterization of these hetero-nanodomains of Li_3−3xAl_xPS₄ and Li₇P₃S₁₁ and their grain boundaries and the demonstration of their effectiveness in solid-state batteries is described in a separate publication.²⁵

2. Experimental

2.1. Synthesis of Al-doped LPS samples

All synthesis procedures were carried out in a high purity argon atmosphere to prevent the material from reacting with moisture and oxygen. The starting materials, Al₂S₃ (MACKLIN, ≥99%), P₂S₅ (MACKLIN, ≥99%), Li₂S (Alfa Aesar, 99.9%), were mixed in an Ar-filled glovebox with target ratios tabulated in Table 1. The mixtures were sealed hermetically into a zirconia jar and ball-milled for 24 h at a speed of 450 rpm using a high energy planetary ball-milling apparatus (Fritsch, Pulverizette 7). After ball-milling, the powders were compressed into pellets and then transferred into an air-tight tube furnace. The compressed pellets were sintered at 300 °C for 8 h under Ar atmosphere and then cooled down to room temperature. Samples are identified as LPSXY where X and Y are the percentage of Li₂S and P₂S₅ in the mixture. The amount of Al incorporated in the Li₃PS₄ structure is expressed as x for the phase with a structure formula of Li_3−3xAl_xPS₄.

Table 1 Nomenclature, stoichiometric molar ratio of precursors for Al-doped LPS samples and their room-temperature ionic conductivities

Sample	Li2S	P2S5	Al2S3	σ (mS cm^−1)
LPS7525	75	25	0	0.2
LPS7426	74	26	4/3	0.64
LPS7327	73	27	8/3	3.30
LPS7228	72	28	12/3	13.2
LPS7129	71	29	16/3	1.43
LPS7030	70	30	20/3	0.33
LPS6931	69	31	24/3	0.03
LPS6535	65	35	40/3	0.01

---

2.2. Electrochemical impedance spectroscopy

Electrochemical impedance spectroscopy (EIS) was conducted on a Biologic VMP-300 electrochemical workstation over a frequency range of 7 MHz to 100 mHz with 50 mV voltage perturbation. The samples for solid-state NMR measurements were obtained by grinding the pellets from the same batch into even and fine powders.

2.3. Powder X-ray diffraction measurements

Powder XRD spectra were collected on a Bruker-AXS Micro diffractometer (D8 ADVANCE) using Cu Kα radiation (1.5406 Å). The scanning speed is 0.02° per step within the range of 2θ = 10°–70°.

2.4. Solid-state NMR spectroscopy

Variable-temperature static ⁷Li solid-state NMR spectra were recorded on a Varian VNMRS system operating at a magnetic field strength of 9.4 T (Larmor frequency: 155.46 MHz for ⁷Li) with a 5 mm Gonio static probe. The sample was packed under N₂ atmosphere in a glovebox and sealed air-tightly. Spin–lattice T₁ relaxation (SLR) was measured using saturation-recovery pulse sequence. A locking-field of 38.8 kHz was used for spin–lattice relaxation measurement in the rotating frame. ⁶Li MAS NMR measurements were perform at a magnetic field of 9.4 T (58.86 MHz for ⁶Li) using a 4 mm MAS HXY Chemagnetics probe. The sample was packed into 4 mm zirconia rotors and spun at 5 kHz. Solid-state ²⁷Al and ³¹P NMR experiments were performed using a Varian VNMRS spectrometer operating at 19.96 T (³¹P Larmor frequency of 344.09 MHz, ²⁷Al Larmor frequency of 221.48 MHz). All ²⁷Al and ³¹P MAS NMR spectra were collected using a HXY triple resonance probe with 1.6 mm zirconia MAS rotors. For ²⁷Al NMR measurements, an “Al-free” rotor was used. The rotor was spun at 32 kHz and 20 kHz for ²⁷Al and ³¹P MAS NMR measurement, respectively. To obtain quantitative ²⁷Al NMR spectra, a 2.7 μs pulse length equivalent to a 30 degree flip angle with a recycle delay of 30 s was used. ³¹P MAS NMR spectra were recorded using 90° pulse of 3 μs length and a recycle delay of 50 s. ²⁷Al MQMAS spectra were acquired using a three pulse sequence with a z-filter. The first two hard pulses for triple quantum coherence excitation and conversion are 2.5 μs and 0.8 μs at an RF strength between 120–130 kHz, respectively. The third soft detection pulse was 8 μs in length at an rf field strength of about 10 kHz. ²⁷Al-{³¹P} rotational-echo, double-resonance (REDOR) NMR experiments were performed at a spin rate of 10 kHz. The spin echo pulse sequence was applied on ²⁷Al nuclei. Meanwhile, rotor-synchronized π pulses were applied on the ³¹P nuclei. Signals with (S⁰) and without (S) ³¹P irradiation were recorded with increasing the number (N) of rotor cycles. The signal attenuation is normalized as ΔS/S⁰ = (S⁰ − S)/S⁰. Then, the REDOR build-up curve was obtained through plotting ΔS/S⁰ as a function of dipolar recoupling time (N × T_r, T_r is the duration of one rotor period). ^6/7Li, ²⁷Al and ³¹P chemical shifts are respectively referenced to LiCl solution, AlCl₃ solution and 85% H₃PO₄. All spectra are processed using the ssNake processing software.²⁷

3. Results and discussion

3.1. Ionic conductivity of Al-doped LPS

The ionic conductivities were determined using impedance spectroscopy (Table 1).²⁵ The pristine LPS7525, i.e. β-Li₃PS₄, has an ionic conductivity of 0.2 mS cm⁻¹ which is comparable to the values reported in the literature.^21–23 With increasing Al content, ionic conductivities rise accordingly. A maximum value of 13.2 mS cm⁻¹ is observed for LPS7228. The ionic conductivity drops dramatically when the Al content is further increased, implying that there is an optimal composition for Li ion transport. The Nyquist plots of all the samples are displayed in Fig. S1 (ESI†). A typical Nyquist plot consisting of a semi-circle and a spike is observed for LPS7525. It is hard to separate bulk and grain boundary conductivity. They are distinguishable only when Li ion motions in bulk and grain boundary respond at different frequencies. In our cases, the overall resistance can be extracted from the EIS measurement. Upon Al incorporation, the overall resistance is reduced drastically in LPS7426. Further Al incorporation leads to liquid-like conductivity in LPS7327 and LPS7228. In order to understand the origin of the enhanced ionic conductivities, we need to gain in-depth insights into the structure and Li ion dynamics at different length scale, i.e. at both macroscopic and microscopic level.

3.2. Structural evolutions in Al-doped LPS

The overall crystal structures of the synthesized Al-doped LPS series were firstly checked with powder X-ray diffraction measurements. The PXRD patterns of LPS samples with different compositions are presented in Fig. S2 (ESI†). Before the Al³⁺ ions are introduced into the host framework, a pure β-Li₃PS₄ is obtained. Upon increasing the Al³⁺ content, no obvious changes are observed in the PXRD patterns of LPS7426. When the Al³⁺ doping level is further increased, several distinct reflections appear in the 2θ range of 20°–25° and 30°–55°, which is indicative of the structural changes. The LPS6931 and LPS6535 samples exhibit a lower crystallinity as evidenced by the significant line broadening of their diffraction lines.

³¹P NMR chemical shifts are used as the indicators to discriminate different polyanionic frameworks in the materials. Fig. S3 (ESI†) displays the 1D ³¹P MAS NMR spectra for samples LPS7525, LPS7228 and LPS7030. Clearly the LPS system undergoes structural evolutions. In the spectrum of LPS7525 (β-Li₃PS₄), only isolated PS₄³⁻ polyhedral at 87 ppm are present.²⁸ When adding Al, P₂S₇⁴⁻ units at 91 ppm appear which are considered to be part of the highly conducting Li7P3S11 electrolyte.²⁹ The appearance of P₂S₇⁴-indicates that there is an upper limit of the Al solubility in β-Li₃PS₄. It is worth noting that a substantial amount of P₂S₆⁴⁻ units (104 and 108 ppm) are found in LPS7030 which could be the reason for the collapse of ionic conductivity at higher Al doping levels.³⁰

To study the local environment of Al in the prepared samples, ²⁷Al MAS NMR spectra were collected as shown in Fig. 2a. On initial inspection we observe a great number of (partially) overlapping resonances. For further analyses we decide to divide the spectrum in 4 spectral regions denoted as I, II, III, IV. Firstly, we have to mention that peaks at around 16 ppm in region IV are due to some oxidized impurities and/or signal from the rotor. Although we used Al-reduced zirconia rotors, there is still a tiny amount of Al present in the rotor material. The ²⁷Al NMR spectrum of an empty Al-reduced rotor is presented in Fig. S4 (ESI†), showing signal at around 16 ppm. The resonance lines above 100 ppm in region I are attributed to 4-fold coordinated Al in AlS₄ polyhedra. This region clearly contains a superposition of several resonances that can be assigned to AlS₄ polyhedra in various polymorphs of Al₂S₃.³¹

Fig. 2 (a) ²⁷Al MAS NMR spectra of Al-doped LPS series, (b) ²⁷Al 3QMAS spectrum of LPS7228. Spinning sidebands were marked with asterisk. Through the ²⁷Al 3QMAS spectrum, we can confirm that the peaks in region I are the superposition of three peaks.

The signals in region IIII and III fall in the regimes of pentahedral and/or octahedral sulphur-coordinated Al sites.^32,33 Based on a comparison with the ²⁷Al NMR spectrum of the precursor, Al₂S₃, it is concluded that the peaks in region I and II are due to Al₂S₃ remaining in the sample, however, the typical reflection peaks for Al₂S₃ are not visible in the XRD patterns. We do observe a broadening of the diffraction peaks, hinting that the Al₂S₃ is significantly distorted as a result of the ball-milling procedure. This was verified using 2D ²⁷Al MQMAS spectroscopy that is able to separate overlapping peaks on the basis of their difference in chemical shift and/or quadrupolar interaction. Fig. 2b presents the MQMAS spectrum of the LPS7228 sample. Obviously, the signals in region I are the summation of multiple peaks which can be well resolved in the isotropic F1 dimension. In addition, we acquired 1D ²⁷Al MAS NMR spectra at different magnetic fields (Fig. S5, ESI†). Going from 14.09 T to 19.96 T, we observe a decreasing line width for the resonances in region I. While the line width of signals in region II and III have a similar line width at both fields indicating that the resonances are affected by both a distribution in chemical shift and the quadrupolar interaction as the chemical shift is proportional to the magnetic field and the second-order inversely proportional, the combination of both can add up to give a similar line width at different fields. The MQMAS spectrum confirms this interpretation.

To establish which aluminium sites have phosphorus in their direct vicinity, ²⁷Al–{³¹P} REDOR experiments were performed on sample LPS7030. Fig. 3a shows the ²⁷Al REDOR spectra with varying ³¹P recoupling time. What can be clearly seen is that only peaks in region III show signal attenuation due to the dipolar recoupling, pointing out the proximity of these Al atoms to P atoms in the structure. Fig. 3b presents the ³¹P magnetization dephasing profile. Since the REDOR fraction, defined as ΔS/S⁰ = 1 − S/S⁰, approaches 1, all these Al are close to P.³⁴ The REDOR spectra of LPS7228 show similar behaviour as shown in Fig. S6 (ESI†). It is interesting that only the Al sites resonating at around 39 ppm are in close proximity to phosphorus atoms. Considering the structure of β-Li₃PS₄ containing isolated PS₄³⁻ polyhedra, the Al ions are most likely substituted in the vacant sites in β-Li₃PS₄ as the octahedral 4b and the tetrahedral 4c sites in the crystal structure are not fully occupied. It is worth to note that the chemical shifts of resonances in region III correspond to aluminium in an octahedral sulphur coordination.

Fig. 3 (a) ²⁷Al (³¹P) REDOR spectra of LPS7030 with increasing rotor periods (T_r), (b) REDOR build-up curve. In ²⁷Al (³¹P) REDOR experiments, the signal attenuation of the ²⁷Al resonance at 39 ppm is caused by the dipolar recoupling to nearby ³¹P. (c) ⁶Li MAS NMR spectra measured at 153 K and 133 K, for LPS7525 and LPS7426, respectively. For LPS7525, we can observe the well-resolved ⁶Li spectrum at 153 K, while for the doped sample, we have to go lower temperature to freeze out Li ion motions since the mobility is enhanced by Al incorporation, (d) Al content in the Al-doped Li_3−3xAl_xPS₄.

To gain insight in the substitution mechanism, ⁶Li MAS NMR experiments were performed on β-Li₃PS₄, LPS7426 and LPS7228, respectively. Room temperature ⁶Li MAS NMR spectra of β-Li₃PS₄ and LPS7228 can be found in Fig. 4. Both spectra are characterized by a single resonance at 0.85 and 0.75 ppm, respectively. In β-Li₃PS₄, three Li sites occur in the structure, namely, Li1 (8d), Li2 (4b) and Li3 (4c), which should occur at a ratio of approximately 20 : 7 : 3 based on the site occupancies.²¹ Stöffler et al. reported a ⁶Li MAS NMR spectrum of β-Li₃PS₄ showing two separate peaks for different Li sites at room temperature.²² In the β-Li₃PS₄ in this study, obtained by mechanosynthesis, all three Li sites are involved in exchange at room temperature or even lower temperature, resulting in a single ⁶Li peak. Due to this fast exchange of Li between these lattice sites on the NMR time scale, only a single exchange-averaged resonance is observed at room temperature. Variable temperature experiments show that site exchange of Li ions over all sites already starts at 173 K (Fig. 4). The difference in the ⁶Li MAS NMR spectrum in this study and the work of Stöffler et al. may be due to the different synthesis procedure.³⁵

Fig. 4 Variable-temperature ⁶Li MAS spectra of (a) β-Li₃PS₄ and (b) LPS7228.

In Fig. 3c the low temperature spectrum of β-Li₃PS₄, the spectrum shows two resonance lines located at 1.14 ppm and 0.56 ppm with an intensity ratio of approximately 2 : 1 (see deconvolution in Fig. S7, ESI†), based on this intensity ratio we assign the 1.14 ppm resonance to Li1 whereas the Li2 and Li3 resonances are thought to overlap at 0.56 ppm. The peak at 1.14 ppm reduces in relative intensity upon increasing the Al content, implying Al incorporation in the lattice leads to a reduction of Li in the Li1 (8d) sites. Combined with the results of the ²⁷Al NMR, we postulate that Al atoms occupy vacant octahedral Li2 (4b) sites, as a result, to maintain charge balance, vacancies are created in the Li1 (8d) sites, hence the reduction in relative intensity of the resonance at 1.14 ppm.

To quantify the incorporation of Al in β-Li₃PS₄, we go back to the 1D ²⁷Al MAS NMR spectra displayed in Fig. 2a. We applied a short 30° pulse to ensure quantitative excitation of all the Al sites in the sample. Table S1 (ESI†) summarizes the integrals of region I, II, III, and IV. Based on that, we can calculate the amount of Al dopants that are doped into the structure in Li_3−3xAl_xPS₄ from the relative ratios of different regions (Fig. 3d). It should be noted again that only Al resonances in region III are from Al in Li_3−3xAl_xPS₄. Based on the intensities of the different regions in the Al spectrum, the maximum amount of Al incorporation in β-Li₃PS₄ is found to be around 0.06, which has also been reported for Al-doped Argyrodite solid electrolytes, probably due to the smaller ionic radius of Al³⁺ (0.535 Å) compared to Li⁺ (0.76 Å).³⁶ Further increase of Al content causes the formation of multiple crystalline and disorder, which is confirmed in the PXRD patterns and ³¹P MAS NMR spectra.²⁵ The formation mechanism of LPS7228 sample was investigated by in situ XRD, cryo-TEM and DFT calculations which is described in a separate paper.²⁵In situ XRD measurements confirms the sequential crystallization process. Li_2.82Al_0.06PS₄ crystallizes initially as the core and Li₇P₃S₁₁ grows on the surface of Li_2.82Al_0.06PS₄ which is observed by TEM. When the Al incorporation level is further increased, in LPS7030 and LPS6931 samples, there exist huge amount of Li₄P₂S₆ and unreacted Al₂S₃ which cause the collapse in Li ion conductivity.

3.3. Li ion dynamics

Fig. 5a and b display the static ⁷Li spectra of the β-Li₃PS₄ and LPS7426 samples while the spectra for the LPS7327 and LPS7228 can be found in Fig. S8 (ESI†). At first sight, all of the spectra display similar characteristics and trends as the temperature rises. At 143 K, a broad base is seen in the range from −20 to +20 kHz which originates from the satellite transitions of the quadrupolar ⁷Li (I = 3/2) nuclear spins. Since the ⁷Li nucleus is a half-integer spin with relatively small quadrupolar moment, the energy levels of ⁷Li are only perturbed by first-order quadrupolar interactions, i.e. the central transition is hardly affected by quadrupolar couplings but broadened by the dipolar interactions with neighbouring spins. The featureless line shape of the satellite transitions is attributed to a broad distribution of the electric field gradient over all ⁷Li sites in the lattice. On top of the satellites, a dipolar broadened central transition is visible. Upon increasing the temperature, motional narrowing (MN) sets in; the Li ion motions, i.e. site exchange between Li1 and Li2 as well as Li3 sites, speed up and exceed the rigid-lattice linewidth, hence the lines narrow. At room temperature, clear quadrupolar features are visible in the line shape for β-Li₃PS₄ corresponding to a mean quadrupolar coupling constant of ∼35 kHz, whereas these features are smeared out in the doped samples due to the increased structural disorder leading to a distribution in quadrupolar couplings caused by Al incorporation.

Fig. 5 Selected variable-temperature (VT) static ⁷Li spectra of (a), β-Li₃PS₄ and (b), LPS7426. Corresponding linewidth of the central transition extracted from the ⁷Li static spectra for (c) β-Li₃PS₄ and (d) LPS7426, respectively.

The changes in the line shape of the static ⁷Li spectra as a function of temperature are more or less a universal behaviour of fast Li ion conductors.³⁷ The linewidth is extracted from the central line and plotted against temperature. As can be seen in Fig. 5c and d, the linewidth reaches a maximum of ∼4 kHz in the rigid-lattice regime where the Li ion mobility is too slow to affect the linewidth, Δν_rl. As the temperature rises, motional narrowing is clearly observed until the Li ion hopping rates exceed the line width 1/τ_MN ≥ 2πΔν_rl. We estimate the onset temperature (T_onset) of the motional narrowing from the inflections in the curve of the linewidth as a function of temperature. Upon increasing the Al doping, the onset temperature for the motional narrowing process shifts to lower temperature. For instance, the motional narrowing process starts at 166 K for LPS7426, which is attributed to enhanced Li ion exchange between Li1 and Li2/Li3 sites. Then the hopping rates τ_MN⁻¹ can be determined at T_onset since 1/τ_MN = 2πΔν_rl.^7,38–41 It should be noted that the motional narrowing process starts at much lower temperature in LPS7228 than in β-Li₃PS₄, indicating enhanced Li ion mobility. Finally, the linewidth reaches a plateau, where the homonuclear ⁷Li–⁷Li dipolar interaction is averaged out due to the fast Li ion hopping. This motional narrowing process is thermally activated and therefore obeys the Arrhenius equation: τ_MN⁻¹ = τ₀⁻¹ exp(−E_MN/(k_BT)). Empirically, the linewidth evolution can be fitted by the phenomenological equation proposed by Hendrickson and Bray,⁴² from which we can obtain the activation energy for motional narrowing and the rigid-lattice linewidth.

Parameters extracted from the fitting are listed in Table 2. Based on these, the average correlation time for Li ion motions at 298 K is calculated to be 3.5 × 10⁻⁸ s for β-Li₃PS₄. According to the Einstein–Smoluchowski equation, the diffusion coefficient D is related to the jumping distance and jumping rates using the formula D = a²/2dτ, where a is the mean squared displacement and d is the dimensionality.^43,44 In β-Li₃PS₄, we assume that a is the shortest distance over which the Li ion can hop (∼3.15 Å). For 3D diffusion, this would yield a diffusion coefficient D = 4.7 × 10⁻¹³ m² s⁻¹. In general, the ionic conductivity σ_NMR can be estimated using the Nernst–Einstein equation: σ_NMR = (DN_Lie²)/(k_BT),^22,41,45 where N_Li = 1.8 × 10²⁸ m⁻³ is the number of Li ions per unit volume, e = 1.602 × 10⁻¹⁹ C is the elementary charge, k_B = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant and T is the temperature in Kelvin. Bringing these values into the formula, σ_NMR = 0.53 mS cm⁻¹ which is much higher than the experimental value.

Table 2 Parameters derived for the best-fits of motional narrowing process

Compositions	E MN (eV)	Δνrl (Hz)	T onset (K)	τ 0 ^−1 (s^−1)
β-Li3PS4	0.26	4163 ± 26	176 ± 2	7.1 × 10^11
LPS7426	0.22	4200 ± 16	166 ± 2	1.3 × 10^11
LPS7327	0.22	4050 ± 43	151 ± 2	5.5 × 10^11
LPS7228	0.23	4050 ± 54	143 ± 2	3.3 × 10^12

---

Another way of gaining insights into ⁷Li ion mobility is through spin–lattice relaxation time measurements. Variable temperature ⁷Li relaxation rates R₁ and R_1ρ collected for all samples are shown in Fig. 6. The temperature dependence of R₁ and R_1ρ can be expressed by the BPP theory through the following equations:^46,47

where C₁ and C₂ are constants related to the interaction strength such as dipolar and quadrupolar interactions; ω₀ and ω₁ are the Larmor frequency and radio frequency field strength, respectively. τ is the correlation time that follows the Arrhenius law τ = τ₀ exp(E_a/(k_BT)). Plots of log(R₁), log (R_1ρ) vs. 1/T are typically characterized by maxima where ω_(0|(1))τ = 0.616 (0.5), which means the average motional rates of ions are at or close to the Larmor frequency in the lab frame or the rotating frame. In the low temperature flank, i.e. slow motional regime with ω_(0|1)τ ≫ 1. Eqn (1) can be reduced to

Fig. 6 Temperature dependence of ⁷Li relaxation rates in the laboratory frame R₁ (a), and in the rotating frame R_1ρ (b).

In the high temperature flank, i.e. fast motional regime with ω_(0|1)τ ≪ 1, eqn (1) can be simplified to

In this way we can extract the activation energy from the slope of the log(R₁) vs. 1/T curves. Fig. 6a shows the temperature dependence of log(R₁). For β-Li₃PS₄ (LPS7525), two relaxation process are present in the low temperature regime. From 223 K to room temperature (RT, 298 K), log(R₁) increases linearly with increasing temperature. This process, characterized by an activation energy of 0.1 eV, is ascribed to very fast local motions, i.e. vibration of Li ion in the lattice. Above RT, another relaxation process sets in which can be attributed to the local hopping of Li ions between adjacent Li sites. In addition, more Li vacancies and interstitials are formed in this temperature range. The apparent activation energy in this temperature range is 0.19 eV which agrees well with the previously reported values for β-Li₃PS₄.^22,23 In order to gain insights into the relaxation mechanisms, the dipolar (R_1D) and quadrupolar (R_1Q) contributions to the relaxation rates are evaluated using eqn (1), At the maximum (T = 453 K), ω₀τ = 0.616 holds. The R_1D and R_1Q are calculated to be 0.117 s⁻¹ and 1.180 s⁻¹, respectively. The calculation details can be found in the supplementary information. The experimental R₁ is, however, 2.039 s⁻¹, indicating the quadrupolar relaxation dominates the process. There is a substantial discrepancy between the calculated and experimental R₁. Beyond the experimental error, the “background” relaxation including paramagnetic impurities induced relaxation is considered to be responsible for the deviation.^48,49 With increasing Al doping levels, the system relaxes faster, indicative of faster motional processes. Upon Al doping, the maxima of log R₁ are shifted to lower temperature (from 453 K to 413 K), providing evidence that the ion mobility has been enhanced. Besides, pronounced changes in relaxation curve shape are observed. The activation energies from the low temperature flank are decreased from 0.19 eV to 0.13 eV upon Al incorporation. For LPS7228, the relaxation rate curves exhibit very broad maxima, which means multiple motional process are present in parallel. Since the domain size in LPS7228 is small, Li ions exchange extremely fast between different domains, therefore, a single T₁ is observed at each temperature. Compared with the SLR data for Li₇P₃S₁₁ reported in the literature,^50,51 it is possible that the process characterized by T_max at lower temperature represents the second spin reservoir from ⁷Li spins in Li₇P₃S₁₁. At least two motional process were visible in ⁶Li SLR for Li₇P₃S₁₁ reported by Wilkening group.⁵⁰ The LPS7228 sample mainly consists of Li_2.82Al_0.06PS₄ and Li₇P₃S₁₁ which further broadens the maxima of R₁ relaxation curve. It is challenging to resolve different motional process from ⁷Li SLR for each component in LPS7228. However, There is no doubt that all types of Li ion motions in Li_2.82Al_0.06PS₄, Li₇P₃S₁₁ and their grain boundaries can contribute to the relaxation process.

The R_1ρ is employed as a probe of Li ion dynamics in the kHz range that is associated with long-range Li ion diffusion. The measured R_1ρ was deconvoluted into a single exponential part and a stretched exponential part. Fig. 6b displays single exponential part log R_1ρvs. 1000/T. At first glance, it seems that only one motional process is present in β-Li₃PS₄ with an activation barrier of 0.13 eV. Similar activation energies are obtained for LPS7426. It is interesting that a very flat slope appears in LPS7228. An activation energy of 0.07 eV is obtained which was reported for Li₇P₃S₁₁ glass ceramics,⁵⁰ indicating a much lower barrier for Li ion diffusion in LPS7228. The activation energy determined by NMR is much smaller than the activation energy determined by EIS measurement.²⁵ This is typical for fast ion conductors because NMR probes Li ion motion at specific frequencies. EIS measurements, on the other hand, reflect an overall Li ion mobility that includes all types of Li ion motion. In most cases, activation energies determined by NMR spin–lattice relaxation are lower than those determined by EIS measurements. With increasing Al doping, the R_1ρ maxima are shifting towards lower temperature which is in line with the R₁ measurements. It should be noted that the maximum of R_1ρ for LPS7228 is not well pronounced, implying multiple Li ion conduction pathways exist in such multiphasic material.

Li ion diffusion in β-Li₃PS₄ occurs preferably along the b axis through Li2 and Li3 sites. 3D bulk Li ion diffusion is hindered by the limited ion jump between bc planes.⁵² When a small amount of Al³⁺ dopants are introduced into the lattice as is the case for sample LPS7426, vacancies are created at tetrahedral Li1 sites, facilitating Li ion diffusion through the lattice, increasing the ionic conductivity is enhanced by up to a factor of ∼3. Interestingly the activation barrier for Li hopping in the pure β-Li₃PS₄ and the doped Li_3−3xAl_xPS₄ (LPS7426) are similar.

²⁷Al MAS NMR spectra demonstrate that the maximum content of Al incorporation in β-Li₃PS₄ is around x = 0.06. Upon increasing the Al content in the synthesis mixture, the highest ionic conductivity is obtained for LPS7228 which is very heterogeneous according to the ²⁷Al and ³¹P MAS NMR spectra as well as the PXRD pattern. Several factors have to be taken into account in interpreting the enhanced ionic conductivities. On one hand, Al incorporation has introduced more vacancies in the crystal structure and promoted the bulk Li ion diffusion. In addition, multiple phases are formed as a result of the synergistic effects of Al incorporation and a shift in the Li₂S/P₂S₅ ratio in the synthesis mixture. A detailed study of the structural evolution of these complicated multiphasic materials, the pathways for the superior Li ion diffusion and their potential application in all-solid-state batteries is described in a separate paper.²⁵

4. Conclusions

In summary, we have investigated the impact of Al incorporation on β-Li₃PS₄ from both the structural and dynamics perspective. Based on ²⁷Al and ⁶Li NMR spectra, we conclude that the Al atoms are embedded into octahedral Li2 (4b) sites while the vacancies are formed Li1 (8d) sites, facilitating Li ion jumping between bc plane. It is found that the maximum incorporation of Al in β-Li₃PS₄ system is approximately x = 0.06. Upon further increase of Al₂S₃ in the synthesis mixture, multiple phases are formed as confirmed by ³¹P NMR spectra; the Li₂S–P₂S₅–Al₂S₃ system becomes more heterogeneous as is corroborated by both the ²⁷Al NMR spectra and the PXRD experiments. As evidenced by ⁷Li NMR relaxometry, different efficient Li ion conduction pathway exist as is explored in ref. 25.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

H. Qu would like to thank the financial support from China Scholarship Council (CSC No. 201804910640). The Netherlands Organisation for Scientific Research (NWO) is greatly acknowledged for the support of the solid-state NMR facility for advanced materials science which is part of the uNMR-NL ROADMAP facilities (NWO project no. 184.035.002). We thank Dr Jennifer Gomez Badillo for her assistance in the double-resonance experiment. Gerrit Janssen, Hans Janssen, and Ruud Aspers are gratefully acknowledged for their technical support.

Notes and references

1. J. B. Goodenough and P. Singh, Review-Solid Electrolytes in Rechargeable Electrochemical Cells, J. Electrochem. Soc., 2015, 162(14), A2387–A2392.
2. A. Manthiram, X. Yu and S. Wang, Lithium Battery Chemistries Enabled by Solid-State Electrolytes, Nat. Rev. Mater., 2017, 2(4), 16103.
3. J. C. Bachman, S. Muy, A. Grimaud, H.-H. Chang, N. Pour, S. F. Lux, O. Paschos, F. Maglia, S. Lupart, P. Lamp, L. Giordano and Y. Shao-Horn, Inorganic Solid-State Electrolytes for Lithium Batteries: Mechanisms and Properties Governing Ion Conduction, Chem. Rev., 2016, 116(1), 140–162.
4. Z. Zhang, Y. Shao, B. Lotsch, Y.-S. Hu, H. Li, J. Janek, L. F. Nazar, C.-W. Nan, J. Maier, M. Armand and L. Chen, New Horizons for Inorganic Solid State Ion Conductors, Energy Environ. Sci., 2018, 11(8), 1945–1976.
5. R. Kanno, T. Hata, Y. Kawamoto and M. Irie, Synthesis of A New Lithium Ionic Conductor, Thio-LISICON–Lithium Germanium Sulfide System, Solid State Ionics, 2000, 130(1), 97–104.
6. R. Kanno and M. Murayama, Lithium Ionic Conductor Thio-LISICON: The Li₂S–GeS₂–P₂S₅ System, J. Electrochem. Soc., 2001, 148(7), A742.
7. C. Vinod Chandran and P. Heitjans, One – Solid-State NMR Studies of Lithium Ion Dynamics Across Materials Classes, in Annual Reports on NMR Spectroscopy, ed. G. A. Webb, Academic Press, 2016, vol. 89, pp. 1–102.
8. S. Ohno, A. Banik, G. F. Dewald, M. A. Kraft, T. Krauskopf, N. Minafra, P. Till, M. Weiss and W. G. Zeier, Materials Design of Ionic Conductors for Solid State Batteries, Prog, Energy, 2020, 2(2), 022001.
9. D. Liu, W. Zhu, Z. Feng, A. Guerfi, A. Vijh and K. Zaghib, Recent Progress in Sulfide-Based Solid Electrolytes for Li-Ion Batteries, Mater. Sci. Eng., B, 2016, 213, 169–176.
10. R. Kanno, M. Murayama and K. Sakamoto, New Lithium Solid Electrolytes, Thio-LISICON: Materials Design Concept and Application to Solid State Battery, Solid State Ion., 2002, 13–22.
11. N. Kamaya, K. Homma, Y. Yamakawa, M. Hirayama, R. Kanno, M. Yonemura, T. Kamiyama, Y. Kato, S. Hama, K. Kawamoto and A. Mitsui, A Lithium Superionic Conductor, Nat. Mater., 2011, 10(9), 682–686.
12. P. Bron, S. Johansson, K. Zick, J. Schmedt auf der Günne, S. Dehnen and B. Roling, Li₁₀SnP₂S₁₂: An Affordable Lithium Superionic Conductor, J. Am. Chem. Soc., 2013, 135(42), 15694–15697.
13. A. Kuhn, O. Gerbig, C. Zhu, F. Falkenberg, J. Maier and B. V. Lotsch, A New Ultrafast Superionic Li-Conductor: Ion Dynamics in Li₁₁Si₂PS₁₂ and Comparison with Other Tetragonal LGPS-type Electrolytes, Phys. Chem. Chem. Phys., 2014, 16(28), 14669–14674.
14. B. Karasulu, S. P. Emge, M. F. Groh, C. P. Grey and A. J. Morris, Al/Ga-Doped Li₇La₃Zr₂O₁₂ Garnets as Li-Ion Solid-State Battery Electrolytes: Atomistic Insights into Local Coordination Environments and Their Influence on ¹⁷O, ²⁷Al, and ⁷¹Ga NMR Spectra, J. Am. Chem. Soc., 2020, 142(6), 3132–3148.
15. H. Aono, E. Sugimoto, Y. Sadaoka, N. Imanaka and G. Y. Adachi, Ionic Conductivity of the Lithium Titanium Phosphate (Li_1+XM_XTi_2−X(PO₄)₃, M = Al, Sc, Y, and La) Systems, J. Electrochem. Soc., 1989, 136(2), 590–591.
16. J. S. Thokchom, N. Gupta and B. Kumar, Superionic Conductivity in a Lithium Aluminium Germanium Phosphate Glass–Ceramic, J. Electrochem. Soc., 2008, 155(12), A915.
17. S. P. Ong, Y. Mo, W. D. Richards, L. Miara, H. S. Lee and G. Ceder, Phase Stability, Electrochemical Stability and Ionic Conductivity of the Li_10±1MP₂X₁₂ (M = Ge, Si, Sn, Al or P, and X = O, S or Se) Family of Superionic Conductors, Energy Environ. Sci., 2013, 6(1), 148–156.
18. P. Zhou, J. Wang, F. Cheng, F. Li and J. Chen, A Solid Lithium Superionic Conductor Li₁₁AlP₂S₁₂ with a Thio-LISICON Analogous Structure, Chem. Commun., 2016, 52(36), 6091–6094.
19. C. Dietrich, D. A. Weber, S. J. Sedlmaier, S. Indris, S. P. Culver, D. Walter, J. Janek and W. G. Zeier, Lithium Ion Conductivity in Li₂S–P₂S₅ Glasses – Building Units and Local Structure Evolution During the Crystallization of Superionic Conductors Li₃PS₄, Li₇P₃S₁₁ and Li₄P₂S₇, J. Mater. Chem. A, 2017, 5(34), 18111–18119.
20. Ö. U. Kudu, T. Famprikis, B. Fleutot, M.-D. Braida, T. Le Mercier, M. S. Islam and C. Masquelier, A Review of Structural Properties and Synthesis Methods of Solid Electrolyte Materials in the Li₂S–P₂S₅ Binary System, J. Power Sources, 2018, 407, 31–43.
21. M. Tachez, J.-P. Malugani, R. Mercier and G. Robert, Ionic Conductivity of and Phase Transition in Lithium Thiophosphate Li₃PS₄, Solid State Ionics, 1984, 14(3), 181–185.
22. H. Stöffler, T. Zinkevich, M. Yavuz, A. Senyshyn, J. Kulisch, P. Hartmann, T. Adermann, S. Randau, F. H. Richter, J. Janek and S. Indris, Ehrenberg, H. Li⁺-Ion Dynamics in β-Li₃PS₄ Observed by NMR: Local Hopping and Long-Range Transport, J. Phys. Chem. C, 2018, 122(28), 15954–15965.
23. D. Prutsch, B. Gadermaier, H. Brandstätter, V. Pregartner, B. Stanje, D. Wohlmuth, V. Epp, D. Rettenwander, I. Hanzu and H. M. R. Wilkening, Nuclear Spin Relaxation in Nanocrystalline β-Li₃PS₄ Reveals Low-Dimensional Li Diffusion in an Isotropic Matrix, Chem. Mater., 2018, 30(21), 7575–7586.
24. J. H. Kennedy, C. Schaupp, H. Eckert and M. Ribes, Aluminium Substitution in the Glass System 0.33[(1−x)P₂S₅−xAl₂S₃]−0.67Li₂S, Solid State Ionics, 1991, 45(1), 21–27.
25. Y. Wang, H. Qu, B. Liu, X. Li, J. Ju, S. Zhang, J. Ma, C. Li, Z. Hu, C.-K. Chang, H.-S. Sheu, L. Cui, F. Jiang, E. R. H. van Eck, A. P. M. Kentgens, G. Cui and L. Chen, Self-Organized Hetero-Nanodomains Actuating Super Li⁺ Conduction in Glass Ceramics, Nat. Commun., 2023 DOI:10.1038/s41467-023-35982-7.
26. K. Momma and F. Izumi, VESTA 3 for Three-Dimensional Visualization of Crystal, Volumetric and Morphology Data, J. Appl. Crystallogr., 2011, 44(6), 1272–1276.
27. S. G. J. van Meerten, W. M. J. Franssen and A. P. M. Kentgens, ssNake: A Cross-Platform Open-Source NMR Data Processing and Fitting Application, J. Magn. Reson., 2019, 301, 56–66.
28. H. Stöffler, T. Zinkevich, M. Yavuz, A.-L. Hansen, M. Knapp, J. Bednarčík, S. Randau, F. H. Richter, J. Janek, H. Ehrenberg and S. Indris, Amorphous versus Crystalline Li₃PS₄: Local Structural Changes during Synthesis and Li Ion Mobility, J. Phys. Chem. C, 2019, 123(16), 10280–10290.
29. Y. Seino, M. Nakagawa, M. Senga, H. Higuchi, K. Takada and T. Sasaki, Analysis of the Structure and Degree of Crystallisation of 70Li₂S–30P₂S₅ Glass Ceramic, J. Mater. Chem. A, 2015, 3(6), 2756–2761.
30. S. Neuberger, S. P. Culver, H. Eckert, W. G. Zeier and J. Schmedt auf der Günne, Refinement of the Crystal Structure of Li₄P₂S₆ Using NMR Crystallography, Dalton Trans., 2018, 47(33), 11691–11695.
31. S. M. Martin and J. A. Sills, ²⁹Si and ²⁷Al MASS-NMR Studies of Li₂S + Al₂S₃ + SiS₂ Glasses, J. Non-Cryst. Solids, 1991, 135(2), 171–181.
32. A. Lafond, X. Rocquefelte, M. Paris, C. Guillot-Deudon and V. Jouenne, Crystal Chemistry and Electronic Structure of the Photovoltaic Buffer Layer, (In_1xAl_x)₂S₃, Chem. Mater., 2011, 23(23), 5168–5176.
33. M. Haouas, F. Taulelle and C. Martineau, Recent Advances in Application of ²⁷Al NMR Spectroscopy to Materials Science, Prog. Nucl. Magn. Reson. Spectrosc., 2016, 94–95, 11–36.
34. T. Gullion and J. Schaefer, Rotational-Echo Double-Resonance NMR, J. Magn. Reson., 1989, 81(1), 196–200.
35. S. Ohno, T. Bernges, J. Buchheim, M. Duchardt, A.-K. Hatz, M. A. Kraft, H. Kwak, A. L. Santhosha, Z. Liu, N. Minafra, F. Tsuji, A. Sakuda, R. Schlem, S. Xiong, Z. Zhang, P. Adelhelm, H. Chen, A. Hayashi, Y. S. Jung, B. V. Lotsch, B. Roling, N. M. Vargas-Barbosa and W. G. Zeier, How Certain Are the Reported Ionic Conductivities of Thiophosphate-Based Solid Electrolytes? An Interlaboratory Study, ACS Energy Lett., 2020, 5(3), 910–915.
36. P. Adeli, J. D. Bazak, A. Huq, G. R. Goward and L. F. Nazar, Influence of Aliovalent Cation Substitution and Mechanical Compression on Li-Ion Conductivity and Diffusivity in Argyrodite Solid Electrolytes, Chem. Mater., 2021, 33(1), 146–157.
37. R. Bertermann and W. Müller-Warmuth, Universality of NMR Results in LISICON Systems and Other Solid Lithium Conductors, Z. Naturforsch., A: Phys. Sci., 1998, 53(10–11), 863–873.
38. K. Volgmann, V. Epp, J. Langer, B. Stanje, J. Heine, S. Nakhal, M. Lerch, M. Wilkening and P. Heitjans, Solid-State NMR to Study Translational Li Ion Dynamics in Solids with Low-Dimensional Diffusion Pathways, Z. Phys. Chem., 2017, 231(7–8), 1215–1241.
39. V. Epp and M. Wilkening, Li-ion Dynamics in Solids as Seen Via Relaxation NMR, Handbook of Solid State Batteries, 2015, pp. 133–190.
40. P. Heitjans, A. Schirmer and S. Indris, NMR and β-NMR Studies of Diffusion in Interface-Dominated and Disordered Solids, in Diffusion in Condensed Matter: Methods, Materials, Models, ed. P. Heitjans and J. Kärger, Springer Berlin Heidelberg, Berlin, Heidelberg, 2005, pp. 367–415.
41. A. Kuhn, P. Sreeraj, R. Pöttgen, H.-D. Wiemhöfer, M. Wilkening and P. Heitjans, Li Ion Diffusion in the Anode Material Li₁₂Si₇: Ultrafast Quasi-1D Diffusion and Two Distinct Fast 3D Jump Processes Separately Revealed by ⁷Li NMR Relaxometry, J. Am. Chem. Soc., 2011, 133(29), 11018–11021.
42. J. R. Hendrickson and P. J. Bray, A Phenomenological Equation for NMR Motional Narrowing in Solids, J. Magn. Reson., 1973, 9(3), 341–357.
43. M. Uitz, V. Epp, P. Bottke and M. Wilkening, Ion Dynamics in Solid Electrolytes for Lithium Batteries, J. Electroceram., 2017, 38(2), 142–156.
44. P. Heitjans, S. Indris and M. Wilkening, Solid-State Diffusion and NMR, 2005.
45. K. Hayamizu, Y. Terada, K. Kataoka, J. Akimoto and T. Haishi, Relationship Between Li⁺ Diffusion and Ion Conduction for Single-Crystal and Powder Garnet-Type Electrolytes Studied by ⁷Li PGSE NMR Spectroscopy, Phys. Chem. Chem. Phys., 2019, 21(42), 23589–23597.
46. N. Bloembergen, E. M. Purcell and R. V. Pound, Relaxation Effects in Nuclear Magnetic Resonance Absorption, Phys. Rev., 1948, 73(7), 679–712.
47. A. F. McDowell, N. L. Adolphi and C. A. Sholl, Site and Barrier Energy Distributions That Govern the Rate of Hydrogen Motion in Quasicrystalline Ti₄₅Zr₃₈Ni₁₇H_x, J. Phys.: Condens. Matter, 2001, 13(43), 9799–9812.
48. M. Wilkening and P. Heitjans, Li Jump Process in h-Li_0.7TiS₂ Studied by Two-Time ⁷Li Spin-Alignment Echo NMR and Comparison with Results on Two-Dimensional Diffusion From Nuclear Magnetic Relaxation, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 77(2), 024311.
49. P. M. Richards, Nuclear Magnetic Relaxation by Paramagnetic Impurities in Superionic Conductors, Phys. Rev. B: Condens. Matter Mater. Phys., 1978, 18(11), 6358–6371.
50. D. Wohlmuth, V. Epp and M. Wilkening, Fast Li Ion Dynamics in the Solid Electrolyte Li₇P₃S₁₁ as Probed by ^6,7Li NMR Spin–Lattice Relaxation, Chem. Phys. Chem., 2015, 16(12), 2582–2593.
51. C. Yu, S. Ganapathy, E. R. H. van Eck, L. van Eijck, N. de Klerk, E. M. Kelder and M. Wagemaker, Investigation of Li-ion transport in Li₇P₃S₁₁ and solid-state lithium batteries, J. Energy Chem., 2019, 38, 1–7.
52. N. J. J. de Klerk, E. van der Maas and M. Wagemaker, Analysis of Diffusion in Solid-State Electrolytes through MD Simulations, Improvement of the Li-Ion Conductivity in β-Li₃PS₄ as an Example, ACS Appl. Energy Mater., 2018, 1(7), 3230–3242.

---

Footnotes

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2cp04670a

‡ These authors contributed equally.

---