Tracking Ions the Direct Way: Long-Range Li+ Dynamics in the Thio-LISICON Family Li4MCh4 (M = Sn, Ge; Ch = S, Se) as Probed by 7Li NMR Relaxometry and 7Li Spin-Alignment Echo NMR.

Solid electrolytes are key elements for next-generation energy storage systems. To design powerful electrolytes with high ionic conductivity, we need to improve our understanding of the mechanisms that are at the heart of the rapid ion exchange processes in solids. Such an understanding also requires evaluation and testing of methods not routinely used to characterize ion conductors. Here, the ternary Li4MCh4 system (M = Ge, Sn; Ch = Se, S) provides model compounds to study the applicability of 7Li nuclear magnetic resonance (NMR) spin-alignment echo (SAE) spectroscopy to probe slow Li+ exchange processes. Whereas the exact interpretation of conventional spin-lattice relaxation data depends on models, SAE NMR offers a model-independent, direct access to motional correlation rates. Indeed, the jump rates and activation energies deduced from time-domain relaxometry data perfectly agree with results from 7Li SAE NMR. In particular, long-range Li+ diffusion in polycrystalline Li4SnS4 as seen by NMR in a dynamic range covering 6 orders of magnitude is determined by an activation energy of E a = 0.55 eV and a pre-exponential factor of 3 × 1013 s-1. The variation in E a and 1/τ0 is related to the LiCh4 volume that changes within the four Li4MCh4 compounds studied. The corresponding volume of Li4SnS4 seems to be close to optimum for Li+ diffusivity.

Introduction

The diffusive motion of small ions such as H+, Li+, and Na+ as well as F– and O2– plays pivotal roles in many branches of materials science.[1] In particular, the design of powerful solid electrolytes for electrochemical energy storage systems, such as Li+ batteries taking benefit from ceramic electrolytes,[2−6] requires an in-depth understanding, which factors influence hopping rates 1/τ and activation energies Ea of the Li+ ions. Besides other methods,[1] nuclear magnetic resonance (NMR) offers a large set of techniques[1,7−21] that can probe Li+ hopping processes on different length scales and time scales in a variety of materials including amorphous (glassy) and crystalline materials.[14]

While some techniques, such as (pulsed or static) field gradient NMR,[13,22−28] can probe macroscopic, that is, tracer diffusion coefficients, others, such as high-resolution 1D or 2D exchange NMR, are sensitive to site-specific Li+ hopping processes.[9−11,29−33] Time-domain NMR methods,[1,12,15] including especially spin–lattice relaxation techniques,[1,15,34] probe short- and long-range ion dynamics,[7] depending on the temperature range used to sample the diffusion-induced relaxation rates. Here, we used a set of model compounds of the Li4MCh4 (M = Sn, Ge; Ch = S, Se) family[35,36] to apply so-called stimulated echo NMR to probe long-range Li+ ion dynamics. In general, stimulated echo NMR is sensitive to extremely slow dynamic processes;[37−44] thus, it can measure jump rates with values in the kilohertz to sub-hertz range.[43] In some cases, this dynamic range is comparable to those obtained by high-resolution exchange spectroscopy.[41] Stimulated echo NMR is, however, also applicable to amorphous solids.[12,37,45−47] If it is combined with methods being sensitive to motional (or rotational) processes taking place at shorter length scales, in ideal cases the dynamic range accessible by NMR methods can cover up to 10 decades.[43]

In the present contribution, we used the quadrupole nucleus 7Li (spin quantum number I = 3/2) to probe two-time spin-alignment echo (SAE) NMR correlation functions.[41,42,48,49] The decay curves directly contain the Li+ jump rate 1/τ if the jumping ion senses electric quadrupolar field fluctuations during its diffusive motion.[42] In other words, SAE NMR monitors the changes in site-specific quadrupole frequency ωq, (i = 1, 2, 3, ...) experienced by the jumping ion during a time interval called tm. The two-time correlation function S2, which is obtained when the echo amplitude is plotted vs mixing time tm, correlates the sine of the initial phase ωq(t = tm → 0)tp with that at a later time t = tm, that is, with sin(ωq(tm)tp). Within this type of experiment the evolution time tp is kept constant. Provided the spin under investigation visits electrically inequivalent sites ωq,, S2 measures the probability of finding a Li ion at a later time in a position equal to that at the beginning. If we assume that hopping of a Li ion does not influence the electric field gradient to which another Li ion is exposed, the S2 decay curve represents a single-particle correlation function. In many cases, its shape can be parametrized by a stretched exponential function S2 ∝ exp(−(tm/τSAE)) with 0 < n ≤ 1. The upper limit of detectable jump rates is given by approximately 104 s–1; this limit is determined by the (quadrupolar) spin–spin relaxation rate.[42,44] The lower limit, however, is given by the quadrupolar spin–lattice relaxation rate (1/T1q (= 1/T1,SAE)), which is in the order of the longitudinal rate 1/T1.[37,41] This influence causes an additional damping of S2, yielding S2(tp, tm, t) = (A exp[−(tm/τSAE)] + B) × exp[−(tm/T1,SAE)γ] with 0 < γ ≤ 1, and if τSAE significantly differs from T1,SAE, a well-resolved final state amplitude S∞ = A/(A + B), where A and B are constants.[41] In such a case, also a sum of two exponentials can be used to approximate S2(tm), as it is done here.

Originally, Spiess and co-workers developed stimulated (spin-alignment) echo NMR to mainly probe rotational dynamics of spin-1 nuclei, such as 2H.[50−52] Translational deuteron dynamics can be probed as well.[53,54] Later, Tang and Wu used the 9Be nucleus[40,48,55] to extent the technique to other quadrupole nuclei with comparable quadrupole interactions as 2H. The first study on 7Li ion conductors was presented by Böhmer and co-workers.[38] Later, the technique was used to characterize a range of materials classes, also by taking advantage of the spin-1 6Li probe[7,46,56−58] characterized by its much smaller quadrupole moment as compared to that belonging to the 7Li spin.[37,38,45,56,59−61] In our own group, the technique has been applied to study ion dynamics in many promising materials for energy storage and in compounds serving as model substances to study the applicability of the technique to crystalline and amorphous samples.[7] These samples include, for example, the garnet-type materials Li7La3Zr2O12[62] and Li6.5La3Zr1.75Mo0.25O12, ternary oxides such as Li2TiO3,[63] Li7BiO6,[64] and Li4SiO4,[41] the anode material Li4Ti5O12,[65,66] argyrodite-type Li7PSe6,[67] and 2D ion conductors such as Li0.7TiS2[42−44] and Li3N[7,56] as well as the glassy aluminosilicate LiAlSi2O6.[46] Recent studies used 7Li SAE NMR to understand ion dynamics in the most famous solid electrolyte Li10GeP2S12 that is known to show exceptionally high Li ion conductivities at ambient temperature.[68−70]

While 6Li and 7Li SAE NMR use electric quadrupolar fluctuations to probe Li exchange rates, Vogel, Eckert, and co-workers took advantage of chemical shift interactions to probe 109Ag NMR motional correlation functions in different kinds of materials.[58,71−76] As they also probed 109Ag NMR four-time correlation functions,[76] they opened the door to measure dynamic heterogeneities in solids. Meanwhile, Böhmer and co-workers presented similar approaches also for the 7Li nucleus.[77] Also, 23Na stimulated echo NMR data were presented to probe Na+ ion dynamics in glasses[78] and Na2O2.[79]

Here, we apply variable-temperature and mixing-time dependent 7Li SAE NMR to study ion dynamics in Li4SnS4 and its Ge- and Se-substituted analogues, namely Li4GeS4, Li4SnSe4, and Li4GeSe4.[36] Inspired by the class of oxide-based LISICONs[80] (Li superionic conductors) in the thio family, oxygen anions are replaced by the more polarizable sulfur anions. This step often improves the ionic conductivity and allows for less energy consuming heat treatments.[81] Li4GeS4 was first synthesized by Matsushita et al.;[82] in the past few years numerous studies investigated the effect of elemental substitutions on ion dynamics.[36,83−88] In a very recent study[35] we investigated the differences in structure, macroscopic, and microscopic transport in the two-dimensional substitution series Li4MCh4 (M = Sn, Ge; Ch = S, Se). Li4MCh4 crystallizes with orthorhombic symmetry (Pnma (No. 62)); the crystal structure consists of MCh4 tetrahedra, LiCh4 tetrahedra, and LiCh6 octahedra.[36] The four samples differ in Li occupation and thus in their Li sublattices. As resolved by neutron powder diffraction,[35] Li4GeS4 and Li4SnSe4 are isostructural and show two different 4-fold coordinated Li sites (Li1, Li2) and one 6-fold Li site (Li3). This is in contrast to Li4SnS4 and Li4GeSe4 which each show an additional Li site in a distorted octahedral (Li4) and in a distorted tetrahedral pocket (Li2′), respectively. Worth mentioning, our Li4SnS4 sample is characterized by a lower ionic conductivity compared to the sample studied by Kaib et al., which was prepared via dehydration of Li4SnS4·13H2O.[85] Here, the samples were synthesized following solid-state protocols (see below). The preparation conditions seem to have a pivotal influence of the final transport properties.

The combination of different but complementary NMR techniques, including temperature-dependent 7Li NMR line shape measurements, diffusion-induced spin–lattice relaxation NMR experiments, and 7Li SAE NMR spectroscopy, helped us to characterize the dynamic processes taking place in Li4MCh4 and to resolve the fine differences between the four samples. In particular, we will show that echo decay rates are fully consistent with both activation energies and jump rates deduced from 7Li NMR spin-lock relaxation rates measured in the so-called rotating frame of reference.

Experimental Methods

Synthesis and Characterization by Neutron Powder Diffraction and Impedance Measurements

The four polycrystalline samples Li4MCh4 (M = Sn, Ge; Ch = S, Se) investigated were prepared by following solid-state syntheses routes; the exact synthesis conditions can be found elsewhere.[35] The same holds for the structural characterization by neutron powder diffraction; the corresponding data of the samples are presented elsewhere.[35,36] The corresponding diffractograms were collected at the Oak Ridge spallation neutron source (SNS, Oak Ridge National Laboratory) by using the PAC automatic sample changer mounted at the POWGEN diffractometer (BM11-A beamline).[35,36] Li4SnS4 and Li4GeS4 crystallize with Pnma space group and turned out to be phase pure.[36] While Li4GeSe4 shows 0.5% GeSe2, for Li4SnSe4 2.5% Li2Se and 2.3% SnSe2 were found.[35] Impedance data of the same samples as studied here are presented elsewhere; here, we refer our results to these data.[35,36]

7Li NMR Spin–Lattice Relaxation Measurements

For the NMR measurements we fire-sealed the powder samples in Duran glass tubes under vacuum. The tubes protect them from any reaction with air or moisture. To investigate Li+ ion dynamics, we recorded 7Li NMR line shapes and 7Li NMR spin–lattice relaxation (SLR) rates. SLR experiments were performed in both the laboratory (1/T1) and the rotating (1/T1ρ) frame of reference by using a Bruker Avance III spectrometer that was connected to a shimmed cryomagnet with a nominal magnetic field of 7 T, corresponding to a Larmor frequency of ω0/2π = 116 MHz. To record free induction decays and (diffusion-induced) magnetization transients, we used a ceramic high-temperature probe (Bruker), which can cover a temperature range of T = 173–493 K. We used a NiCr-Ni (type K) thermocouple to measure the temperature in the sample chamber. The temperature was monitored and adjusted with a Eurotherm temperature controller and a BVT-3000 heating unit (Bruker) connected to a heater placed well below the sample chamber. The heater was operated with a stream of dry N2 gas running through a Dewar filled with liquid N2 to reach temperatures below ambient.

Laboratory-frame 7Li NMR SLR rates were sampled with the well-known saturation recovery sequence.[89] A comb of 10 closely spaced 90° pulses destroys any longitudinal magnetization M; its recovery due to diffusive motions or other sources of spin fluctuations is then recorded with a 90° detection pulse as a function of waiting time td. At a power level of 200 W the 90° (π/2) pulse length varied from 2.13 to 2.83 μs depending on temperature. To construct M(td), the area under the free induction decays (FIDs) was plotted vs td. We accumulated up to 16 scans per td to obtain each FID; altogether, M(td) is composed of up to 18 data points. In general, the magnetization curves were analyzed with stretched exponentials, M(td) ∝ 1 – exp(−(td/T1)γ), yielding the exponent γ and the 1/T1(1/T) spin–lattice relaxation rate; γ = 1 is obtained for pure exponential transients. Only for Li4SnS4 (ca. 300–400 K) the transients show biexponential behavior (see below), and we used a sum of two stretched exponentials to parametrize the curves; the corresponding rates differ by more than 1 order of magnitude.

The 7Li NMR spin–lattice relaxation rates in the rotating-frame of reference[90] were recorded with the spin-lock pulse sequence at a (technical) locking frequency of ω1/2π = 20 kHz. To meet the same locking conditions (20 kHz) at each temperature, the pulse power of the locking pulse was varied from 5.1 to 18.5 W. At the beginning of the spin-lock experiment, a 90° pulse flips the equilibrium magnetization M into the (xy)′-plane of the rotating frame of reference. Directly after, the variable spin-lock pulse is sent that forces M to adapt to the new magnetic field. FIDs were directly recorded after the spin-lock pulse. We used up to eight scans per locking time tlock to record one FID(tlock). A recycle delay of 5T1 ensured full longitudinal relaxation between each scan. Again, up to 18 data points were used to construct the transversal decay curve Mρ(tlock) ∝ exp(−(tlock/T1ρ)γ′), with γ′ = γρ, by plotting the area under the accumulated FIDs vs tlock. As in the case of 1/T1, the transients Mρ(tlock) were, if not stated otherwise, evaluated with stretched exponential functions.

7Li Spin-Alignment Echo NMR

7Li NMR SAE decay curves were recorded on the same spectrometer as used for the 1/T1 measurements (Bruker Avance III (7 T)); our measurements covered a temperature range from −35 to 30 °C. To generate stimulated, that is, spin-alignment, echoes, we employed the three-pulse sequence that was introduced by Jeener and Broekaert to create quadrupolar order[49,91] (π/2)–tp–(π/4)–tm–(π/4)ϕ–tp–echo (acquisition); the 90° pulse length varied from 2.22 to 2.67 μs. A short preparation time of only 15 μs ensured the formation of a quadrupolar spin-alignment state; dipolar contributions were almost suppressed under these conditions as the corresponding echoes consist of a sharply decaying quadrupole part and a slowly decaying dipolar contribution. We used up to 18 different mixing times to construct the decay curve S2(tp = 15 μs, tm); for each echo up to 32 signals were accumulated. A 32-fold phase cycling was used to suppress unwanted coherences.[45] Between each scan a recycle delay of 5T1 was ensured. Depending on temperature, the mixing time was varied from 10 μs to 158 s. The echo decay curves, that is, the two-time correlation functions, were parametrized with a combination of two stretched exponentials as S2, in the T range covered here, reflects both the loss of phase coherence because of Li+ jump processes (1/τSAE) and the decay of S2 because of ordinary (quadrupolar) spin–lattice relaxation (1/T1,SAE) (see above). The latter damping process, however, appears at longer mixing times and can be clearly separated from the main SAE NMR decay step as the corresponding decay rates differ by several orders of magnitude. Further details on the 6,7Li SAE NMR technique, its principle, advantages, and limitations are presented and discussed elsewhere.[12,20,48,49]

6,7Li Magic Angle Spinning (MAS) NMR

6,7Li MAS NMR spectra were recorded with a Bruker 500 MHz spectrometer that is connected to a shimmed Bruker 11.7 T magnet (6Li (73.6 MHz); 7Li (194.4 Hz)). We used a standard Bruker probe that can spin 2.5 mm rotors at 25 kHz. The rotors were filled under an Ar atmosphere. Spectra were acquired at 30 °C with 128 and 64 scans. At a power level of 100 W the 6Li 90° pulse length was 4.5 μs (recycle delay 10–40 s); for 7Li (180 W) the pulse length was 1.4 μs (recycle delay 1 s). Spectra were referenced to solid lithium acetate (1 M).

Results and Discussion

Motion-Induced Changes of 7Li NMR Spectra

The investigation of static NMR (central) lines recorded at different temperatures entails information about ion dynamics that can average dipolar Li–Li couplings.[92] In Figure a, as an example, the variable-temperature 7Li NMR spectra of Li4SnS4 are shown. The Gaussian-shaped central line seen at low T, that is, the temperature range called the rigid-lattice regime, evolves into a Lorentzian-shaped one. This process turned out to be rather homogeneous, and only at intermediate temperatures (253–283 K) does a two-component line shape appear; the same behavior is seen for Li4GeS4. Likely, this feature results from two spin reservoirs slightly differing in ionic mobility.[93] It is, however, absent for the two selenides investigated.

Figure 1

(a) Variable-temperature 7Li NMR lines of Li4SnS4 (170–433 K). (b) 7Li NMR spectra of Li4MCh4 recorded at 273 K (the values represent the full width at half-maximum, fwhm). Magnifications show the quadrupole part of the spectra at 433 K, and coupling constants are in the order of 30 kHz (M = Ge) as well as 16.5 kHz (Li4SnS4) and 20 kHz (Li4SnSe4). For Li4SnS4 and especially for Li4GeS4 we notice a two-component line shape of the central line at intermediate temperatures, that is, at around 273 K. The narrow line on top of the broad one reflects Li+ ions belonging to dynamically distinct ensembles. (c) Stacked plot of the motional narrowing (MN) curves. The scaling refers to that of Li4SnSe4; dashed and solid lines show fits according to the models of Hendrickson and Bray as well as according to Abragam (see key). Values indicate the onset temperatures, which were determined via the tangential lines. Inset: MN curves (Abragam fits) without any additional offset to evaluate the rigid-lattice line widths. (d) 7Li and 6Li MAS NMR lines of Li4MCh4 recorded at ambient bearing gas (25 kHz spinning speed). Values show the widths of the lines in Hz.

In addition, careful inspection of the lines, which were obtained via excitations with single pulses, reveals broadened quadrupole intensities that can be approximated, in the case of Li4SnS4, with a Gaussian function. At higher T, a motionally averaged quadrupolar powder pattern emerges with the 90° singularities appearing at approximately ±8.25 kHz (see arrows in Figure a,b). Figure b shows spectra recorded at 273 K and magnifications of the quadrupole intensities seen at 433 K. Assuming a mean electric field gradient with axial symmetry, the corresponding coupling constant Cq is given by 16.5 kHz.

Almost the same behavior is seen for the Li4SnSe4 sample. At low temperatures, the corresponding quadrupole pattern shows distinct singularities at ±10 kHz, leading to Cq = 20 kHz; in contrast to Li4SnS4 these are already seen at 313 K at ±10 kHz (see Figure S1). At temperatures as low as 253 K relatively well-defined singularities are also observed for Li4GeS4 and Li4GeSe4 (see Figure S1): the corresponding coupling constants are in the order of 30 kHz (Li4GeSe4) and 40 kHz (Li4GeS4), respectively. The most prominent change in the shape of the quadrupole powder pattern is seen for the Li4GeS4 sample when increasing the temperature from 253 K (Cq ≈ 40 kHz) to 433 K (Cq = 30 kHz) (see Figure S1). This behavior is more or less in line with what is known as the universal NMR response for materials with Li occupying several electrically inequivalent Li sites, as discussed by Müller-Warmuth and co-workers.[94−96]

By reading off the width of the central lines and by plotting the full width at half-maximum as a function of temperature, we constructed the so-called motional narrowing curves shown in Figure c. Figure c uses a stacked plot to compare the shape and onset temperatures Tonset of the individual curves. For Li4SnS4 and Li4GeS4 the beginning of motional narrowing is slightly shifted toward lower temperatures by Δ = 10–15 K with regard to those of the corresponding selenides. This shift indicates that in the sulfides ionic motion kicks in somewhat earlier. At ca. 350 K the four curves have reached the extreme narrowing regime. The residual line width is mainly governed by homogeneities of the external magnetic field (see also the inset of Figure c).[97] The rigid-lattice line width reflects the strength of Li–Li dipolar interactions, which turned out to be strongest for Li4SnS4. Its motional narrowing curve perfectly agrees with that reported very recently by Kwak et al.[84]

Here, we analyzed the motional narrowing curves shown in Figure c with the phenomenological expression introduced by Hendrickson and Bray (dashed lines in Figure c)[98,99] and with the model derived by Bloembergen et al.[100] and Abragam[101] (see solid lines). As expected, the analysis according to Hendrickson and Bray yielded somewhat larger activation energies Ea than that obtained by using the Abragam formalism (see Table ). Hence, better agreement is seen between Hendrickson–Bray activation energies and those from other NMR techniques being sensitive to long-range ion dynamics. Note that the Abragam formalism was developed for spin systems with simple geometries and does not consider complex Li dynamics as well as the superposition of different processes contributing to overall line narrowing. For this reason, activation energies extracted from narrowing curves are to be regarded as estimates. Importantly, both methods yield very similar trends in Ea; for Ch = S we obtain lower activation energies than for the two Se-bearing samples.

Table 1

Activation Energies Ea as Deduced from the Analysis of Motional Line Narrowing According to the Models Introduced by Hendrickson and Bray (HB) as Well as by Abragam (A)a

sample	Ea,A (eV)	Ea,HB (eV)	Ea,ρ (eV)	Ea,BPP (eV)	Ea,SAE (eV)	Ea,1 (eV)
Li4SnS4	0.33(3)	0.52(4)	0.55(3)	0.54(2)	0.55(2)	0.38(4)b
Li4GeS4	0.28(3)	0.43(4)	0.56(3)	0.55(4)c	0.56(2)	0.17(1)d
Li4SnSe4	0.43(4)	0.70(5)	0.69(2)	0.69(2)	0.72(1)	0.15(2)e
Li4GeSe4	0.41(4)	0.68(5)	0.58(3)	0.62(2)	0.58(3)	0.25(1)

Ea,BPP represents activation energies from diffusion-induced spin-lock rate peaks 1/T1ρ(1/T), and Ea,ρ refers to the energy simply obtained by analyzing the linear part of the low-T flank of those peaks, that is, in the regime ω1τ ≪ 1. Ea,SAE denotes activation energies from spin-alignment echo NMR, while Ea,1 corresponds to activation energies from 7Li NMR spin–lattice relaxation measurements carried out in the laboratory frame of reference. The latter characterize short-range ion dynamics as the flanks belong to the limit for which ω0τ ≪ 1 holds.

Biexponential behavior of the magnetization transients; the value refers to the rates 1/T1,fast.

Li4GeS4 shows two rate peaks with almost the same activation energies; the peak at higher T is characterized by 0.53(6) eV.

At higher temperatures the flank is characterized by 0.18(2) eV.

At higher temperatures the slope of the flank increases and points to 0.41(4) eV.

Considering the temperature range of ΔMN = 60 K covered by the motional narrowing curves, it starts at 260 K and ends at 320 K (Li4SnS4). We expect no extremely large distribution of jump rates in the selenides and sulfides, which seems to be in contrast to the sample studied by Kaib et al. earlier. Finally, we also used the Waugh–Fedin expression[102] to estimate activation energies. This expression transforms Tonset into an activation energy according to Ea/eV ≈ (1.617 × 10–3) × Tonset/K and does not take into account the shape of the MN curve. In many cases, especially if heterogeneous dynamics is present, it underestimates the activation energy. For Li4MCh4 the following values are obtained: 0.42 eV (Li4SnS4, Li4GeS4) and 0.44 eV (Li4GeSe4, Li4SnSe4). For the selenides the values are in surprisingly good agreement with those estimated by the Hendrickson–Bray ansatz.

To study whether the structural details of the four compounds are mirrored by 6,7Li MAS NMR spectroscopy,[103] we recorded spectra at 303 K at a spinning speed of 25 kHz (see Figure d). In agreement with the larger electric field gradients and, thus, coupling constants Cq of the Ge-containing samples, the corresponding 7Li MAS NMR lines are somewhat broader (ca. 141 and 174 Hz) than those of the Sn-bearing samples (195 and 214 Hz). Although first-order quadrupole interactions are eliminated by (simple) MAS, second-order effects still affect the 7Li MAS NMR lines slightly. Hence, 6Li is the nucleus of choice to record high-resolution NMR spectra. Here, because of the (i) weaker 6Li–6Li dipolar couplings[104] and (ii) the smaller quadrupole moment q of the spin-1 nucleus (q(6Li) = q(7Li)/50)), almost unperturbed lines were observed for 6Li. The corresponding 6Li MAS NMR line widths range from 18 to 21 Hz. Any spinning sideband manifolds arising from the quadrupole transitions are absent. In general, they appear at integer multiples of the spinning frequency. Their absence is expected because at ambient temperature the Li+ jump rate reaches values in the order of 105 s–1. This rate is sufficiently high to fully average the anisotropic quadrupole interactions.

Importantly, except for Li4SnS4 single lines are seen. A split line, as it is observed for Li4SnS4 (Figure d), has already been observed in the literature[84,85] at 206 K and at lower spinning speeds. Kaib et al.[85] assigned the less intense signal to the octahedrally coordinated Li ions, while the main signal is meant to represent the tetrahedrally coordinated Li ions. Li4GeS4, Li4SnSe4, and Li4GeSe4 have also octahedrally coordinated Li sites;[35] the corresponding signal is, however, absent in our 6Li MAS NMR spectra. As shown by neutron diffraction, and in contrast to the other samples, in Li4SnS4 the Li+ ions occupy the Li4 site (Wyckoff position 4c) in a unique fashion. We suggest that this extra Li site is at least partly causing the additional line near 1 ppm seen in the 6Li MAS NMR spectrum; a deconvolution of the 6Li MAS NMR line is shown in Figure S2. The distribution of Li+ over several crystallographic sites in Li4SnS4 also significantly impacts Li+ ion dynamics in Li4SnS4 (vide infra). In terms of isotropic chemical shifts, the following trends are observed. Increasing the lattice spacing through the replacement of S by Se shifts the 6Li MAS line toward negative ppm values. The same shift is seen when Ge is substituted for Sn.

Diffusion-Induced 7Li NMR Spin–Lattice Relaxation Rates

7Li NMR spin–lattice relaxation experiments were performed in both the laboratory and the rotating frame of reference not only to collect information about activation energies of short- and long-range ion dynamics of the underlying motional processes but also to probe Li+ jump rates.[8,105] The corresponding Arrhenius plots of 1/T1(ρ)(1/T) for each sample are shown in Figure a–d. In the upper graphs, the change of the stretching exponents γ(ρ) is displayed to compare their temperature behavior with those of the corresponding rates 1/T1(ρ).

Figure 2

Arrhenius plot of the 7Li NMR spin–lattice relaxation rates 1/T1 (squares, 116 MHz) and the spin-lock rates 1/T1ρ (circles, 20 kHz) of (a) Li4SnSe4, (b) Li4SnS4, (c) Li4GeSe4, and (d) Li4GeS4. Solid lines, used to parametrize the temperature behavior of the rotating-frame 1/T1ρ rates, represent fits with BPP-type Lorentzian-shaped spectral density functions; for Li4GeS4 the overall 1/T1ρ response is a superposition of two rate peaks (see dotted lines). Dashed lines are drawn to guide the eye; dotted lines refer to the samples indicated, which have been included for better comparisons. Activation energies, Ea,BPP, marked with asterisks (∗) were obtained from the analysis with BPP fits with the respective maxima Tmax indicated. Ea,BPP agree with those if the slopes of the respective peaks in the high-T limits are analyzed with linear fits. Activation energies, Ea,1, belonging to the partly linear regions of the 1/T1 rates and those associated with the low-T flanks of the 1/T1ρ(1/T) rate peaks, Ea,ρ, are also included (see also Table ). The upper graphs illustrates the temperature behavior of the corresponding stretching exponents γ(ρ); lines are to guide the eye. See text for further details.

Starting with the Sn-containing samples, in the limit ω0τ ≫ 1 we see that for Li4SnSe4 the rate 1/T1 reveals only a shallow temperature dependence up to T = 333 K (Figure a). Below 250 K the so-called nondiffusive background relaxation appears,[93] which is mainly due to lattice vibrations and coupling of the Li spins to paramagnetic impurities.[105] The flanks seen at higher T, still in the regime for which ω0τ ≫ 1, are to be characterized by activation energies of 0.15 and 0.41 eV, respectively (see also Figure a and Table ). We assume that below 333 K localized and/or correlated forward–backward Li+ jump processes control 1/T1.

Whereas 1/T1 NMR is, in general, sensitive to spin fluctuations with rates on the MHz range (ω0/2π = 116 MHz), spin lock 1/T1ρ NMR can probe hopping processes with motional correlation rates in the kHz range (ω1/2π = 20 kHz). Here, via variable-temperature 1/T1ρ7Li NMR, we were able to record the complete diffusion-induced rate peak[7,19,106] whose maximum is located at Tmax = 328 K (see Table ). The maximum in 1/T1ρ corresponds to a minimum in γρ.

Table 2

Activation Energies Ea,BPP and Arrhenius Prefactors 1/τ0 Obtained from Analyzing the 7Li NMR 1/T1ρ(1/T) Rate Peaks with the Help of the BPP-Type Spectral Density Function (See Figure )a

sample	Ea,BPP (eV)	1/τ0 (s–1)	Tmax (K)	C (s–2)	β
Li4SnSe4b	0.69(2)	7(5) × 1015	328	9.6(9) × 108	2
Li4SnS4	0.54(2)	3(2) × 1013	328	1.4(2) × 109	2
Li4GeSe4b	0.62(2)	1.1(8) × 1015	320	7.0(7) × 108	2
Li4GeS4c	0.55(4)	5(8) × 1013	320	1.9(6) × 108	2
	0.53(6)	4(7) × 1012	360	3.3(6) × 108	2

C denotes the overall coupling constant; the parameter β = 2 was used to check any deviations from symmetric behavior of the peaks. Tmax refers to the temperatures at which the respective peak occurs.

Samples with high activation energies (0.69 and 0.62 eV) are also characterized by large Arrhenius prefactors reaching values as high as 1015 s–1; an observation being in line with the so-called Meyer–Neldel compensation rule.

The Li4GeS4 sample shows two rate peaks with quite similar activation energies but different prefactors.

In general, 1/T1(ρ) is directly proportional to the spectral density function J(ω0), which is the Fourier transform of the motional correlation function G(t′) representing the spin fluctuations.[1] The NMR rate peak 1/T1ρ of Li4SnSe4 can be well parametrized with a Lorentzian-shaped spectral density function J(ω0(1)) = C × τc/(1 + (ω0(1)τc)β) based on the model of Bloembergen, Purcell, and Pound (BPP).[100] 1/τc denotes the motional correlation rate that usually follows Arrhenius behavior, 1/τc = 1/τc0 × exp(−Ea/(kBT)). The parameters determining the BPP fits are summarized in Table . C represents the overall coupling constant, which governs the amplitude of the rate peaks. To take into account background effects, we added a power law term, 1/T1ρ ∝ Tκ,[42] to approximate the deviations at low (and again at high) temperatures from the 1/T1ρ ∝ J(ω0(1)) behavior; as an example, these deviations are best seen for the Li4GeSe4 sample (see Figure c).

Within a factor of 2 the rate 1/τc equals the Li+ jump rate; 1/τc0 represents the prefactor that mirrors the attempt frequency of the hopping process under investigation. This formalism shows that when 1/T1ρ(1/T) reaches its maximum value at T = Tmax, the mean correlation rate is on the order of the locking frequency used to sample the NMR rate. For spin-lock NMR, the condition ω1τ ≈ 0.5 holds[105,107] at Tmax; hence, if we insert a locking frequency of ω1/2π = 20 kHz, the jump rate 1/τ (Tmax) is approximately given by 2.51 × 105 s–1. Here, we do not consider any local magnetic fields that might lead to an effective frequency ωeff > ω1. Compared to other Li-ion conductors showing jumps rates on the order of 109 s–1 at similar temperatures,[108] Li4SnSe4 with an BPP activation energy of 0.69 eV turned out to be a moderate ion conductor with relatively low motional correlation rates.

The parameter β describes the deviation[108] from a symmetric peak shape, which is obtained for β = 2. Here, the rate peak of Li4SnSe4 shown in Figure a is fully symmetric in shape and points to 3D uncorrelated Li motion as assumed by the (isotropic) BPP model. A very similar 7Li NMR rate peak is also seen for Li4SnS4; the symmetric BPP peak is, however, characterized by a considerably lower activation energy of 0.54 eV (Figure b). The fact that Ea differs but the two rate peaks appear at almost the same temperature Tmax (see also Table ) reveals that the prefactor 1/τc0 of the Arrhenius line characterizing Li ion dynamics in Li4SnS4 is lower than that of the Se counterpart (Table ). Figure a helps to illustrate this situation. Indeed, the BPP fits yield that 1/τc0(Li4SnS4) = 3(2) × 1013 s–1 is by 2 orders of magnitude lower than that belonging to Li4SnSe4 (1/τc0(Li4SnSe4) = 7(5) × 1015 s–1; see Table ). Thus, the underlying Arrhenius lines of 1/τc of the two samples differ in both Ea and 1/τc0 but cross at Tmax ≈ 330 K. Here, at a locking frequency of ω1/2π = 20 kHz the intercept point of the two Arrhenius lines is coincidently probed by spin-lock NMR. Figure a also shows the dynamic range covered by 7Li NMR. For the present investigation, the determination of 1/T1ρ(1/T) alone covers a frequency range of 5 orders of magnitude, which is extended by 7Li SAE NMR measurements (vide infra).

Figure 3

(a) Arrhenius lines for Li4SnS4 and Li4SnSe4 illustrating the temperature range and the dynamic region probed by 7Li spin-lock NMR at 20 kHz. The lines are governed by both different activation energies Ea,BPP and prefactors 1/τ0. The lines cross at ca. 330 K. Inset: Arrhenius lines of the two superimposed rate peaks of Li4GeS4 (see Figure d). (b) 7Li SAE NMR two-time correlation functions S2(tp = 15 μs, tm) of Li4SnS4 and Li4SnSe4 recorded at 273 K and 116 MHz. The values show the stretching factors n of the exponential decay (solid lines). Inset: S2 curves of Li4GeSe4 and Li4GeS4. (c) 7Li SAE NMR decay rates 1/τSAE and (quadrupolar) spin–lattice relaxation rates 1/T1,SAE as extracted from the (biexponential) S2 curves. (d) Arrhenius plot to compare the activation energies of Li4SnS4 from 1/T1 NMR (0.38 eV), 1/T1ρ NMR (0.55 eV), and SAE NMR (0.55 eV). The jump rate 1/τρ deduced from the 1/T1ρ peak perfectly agrees with 1/τSAE(1/T) if extrapolated toward 330 K.

The lower activation energy extracted for Li4SnS4 is also supported by impedance measurements performed on the same samples; they yielded a mean activation energy of ∼0.47 eV and a conductivity of ca. 2 × 10–6 S cm–1 at ambient conditions.[35,36] For the discussion with conductivity results from Kaib et al.,[86] who used the wet-chemical different preparation route to obtain Li4SnS4, we refer to the literature.[36] Here, the presence of faster Li+ exchange processes in Li4SnS4, which seem to trigger long-range transport, is additionally underpinned by 1/T1 NMR. In contrast to Li4SnSe4, for Li4SnS4 a shallow 1/T1 peak appears slightly above 333 K (see the corresponding arrow in Figure b). In this temperature region, the magnetization transients clearly show biexponential behavior characterized by the rates 1/T1,fast and 1/T1,slow which differ by more than 1 order of magnitude; in such a case a single, stretched exponential is no longer useful to parametrize the full transient. The rate 1/T1,fast passes through a rate peak whose amplitude suffers from relatively strong background relaxation. This rate peak, governed by an activation energy of 0.38 eV (Table ), is in line with the measurements reported by Kwak et al.[84] Most likely, it is caused by rapid, but short-ranged, that is, localized (forward–backward) spin fluctuations with characteristic rates on the megahertz time scale. As this important feature is missing for Li4SnSe4, it might reflect a subset of Li spins that benefit from the structural properties being specific for Li4SnS4.[35,36]

Worth noting, the shallow 1/T1 NMR peak seen for Li4SnS4 does not affect spin-lock relaxation as the corresponding 1/T1ρ NMR peak, being expected to appear at sufficiently low temperatures, is missing. Obviously, the two methods are sensitive to quite different spin fluctuations in Li4SnS4, which is in contrast to impedance spectroscopy probing average dynamic properties that characterize long-range ionic transport.[35] We assume that if several Li ion jump processes run in parallel, not all of them will contribute to the distinct, time-scale-dependent NMR responses in equal shares. The process characterized by 0.38 eV (Figure b) is, however, considered as a key feature to explain the enhanced transport properties observed for Li4SnS4 by impedance spectroscopy,[35] as it is more pronounced in Li4SnS4 than in the other samples studied.

Continuing with the Ge samples and considering the results from recent impedance measurements on the same samples,[35] we expect that Li+ dynamics at room temperature should turn out to be faster in Li4GeSe4 than in the sulfide Li4GeS4 (see Figure c,d). Indeed, the almost symmetric spin-lock NMR peak 1/T1ρ(1/T) of Li4GeSe4 appears at Tmax = 320 K, that is, even slightly lower than that of Li4SnS4. The full 7Li NMR response is to be characterized by a relatively high BPP activation energy, which corresponds to that of the high-T flank (ω1τ ≪ 1) of the peak (0.62 eV, see Table ). Remarkably, as for Li4SnS4 the laboratory-frame 1/T1 rate of Li4GeSe4 pass through a shallow maximum located at Tmax ≈ 290 K (Figure c and Table ). Again, it is likely that in site-disordered Li4GeSe4 with the additional Li position Li2′ the rapid, partly localized, Li+ diffusion processes probed by 1/T1 NMR also influence long-range ionic transport measured by impedance spectroscopy.[35] These processes could also serve as an argument to explain the lower activation energy compared to Li4GeS4.

The Li4GeS4 sample does also reveal a shallow but broad 1/T1 peak, whose maximum is shifted toward higher temperatures (330 K, see Figure d), clearly revealing a slower diffusion processes than that seen for Li4GeSe4. The spin-lock 1/T1ρ7Li NMR rate peak appears at 320 K; its activation energy on the low-T side is lower but comparable to that probed for the Se counterpart (Table ). Apart from these similarities, the most important difference is revealed by the second 1/T1ρ peak of Li4GeS4, which is visible at a much higher temperature Tmax = 360 K (see Figure d and Table ). Obviously, in Li4GeS4 the Li+ diffusivity is stepwise activated as the 1/T1ρ rate consecutively pass through two different rate peaks (see the dashed lines in Figure d; the solid line is the sum of the two subpeaks). Compared to Li4GeSe4 (1/τc0 = 1.1 × 1015 s–1) the peak of Li4GeS4 seen at higher T is characterized by a significantly lower prefactor 1/τc0 = 4 × 1012 s–1 (see Table for the corresponding BPP fit parameters). Likely, the densely packed structure of Li4GeS4 with the smallest volume of the Li-Ch tetrahedrons among the four materials[35] is responsible for the 1/T1ρ peak being evident at high temperatures.

Li Jump Rates as Deduced from 7Li SAE NMR

Whereas 1/T1ρ provides direct, that is, a model independent, access to Li jump rates only at T = Tmax, the S2 echo decay curves of spin-alignment echo NMR allow for a direct measurement of the underlying motional correlation function.[7,43,47] For this purpose we recorded sine–sine, single-spin two-time correlation functions S2 at constant evolution time tp = 15 μs but variable mixing time tm. The echo formed after the (π/4)ϕ reading pulse is composed of a sharply decaying quadrupole part and a slowly decaying dipolar signal. By plotting the echo amplitude S2 against tm, we obtained decay curves that follow biexponential behavior. Exemplarily, the S2(tp = 15 μs, tm) curves of Li4SnS4 and Li4SnSe4 measured at T = 273 K are shown in Figure b; a full set of temperature-dependent S2 curves of Li4SnS4 is presented in Figure S3. The inset of Figure b displays the corresponding curves of the Ge-bearing samples.

Solid lines in Figure b show fits with a sum of two stretched exponentials. At long mixing times the echo decay is determined by the rate 1/T1,SAE, being in the same order of magnitude as 1/T1[41] (see Figure d and Figure S3). The first decay step, characterized by the rate 1/τSAE, directly reflects Li+ jump processes between the electrically inequivalent sites in the sulfides and selenides that participate in ion dynamics. Accordingly, SAE NMR represents a straightforward way to access hopping rates 1/τ (= 1/τSAE) because no models are needed to convert the primary observable into dynamic parameters, as it is usually the case for spin–lattice relaxation measurements.

In Figure c, the rates 1/τSAE, recorded from 240 to 303 K, are analyzed by using an Arrhenius graph, which also includes the rates 1/T1,SAE for comparison. 1/τSAE(Li4SnS4) follows linear behavior; the Arrhenius fit yields an activation energy of Ea,SAE = 0.55(2) eV, which is in superb agreement with that obtained from 1/T1ρ NMR (0.54(2) eV) (see Table ). This similarity also holds for the other samples whose 7Li SAE NMR decay rates are included in Figure c as well (see Table ). The perfect agreement between the activation energies of the two techniques (Ea,SAE = Ea,BPP) is in detail illustrated in Figure d by using the data for Li4SnS4. If we insert the jump rate deduced from the corresponding 1/T1ρ(1/T) peak, it is unveiled that also the absolute Li+ jump rate measured by spin-lock NMR (1/τ330 K = 2.5 × 105 s–1) is in excellent agreement with the SAE NMR motional correlation rates. The proof that 1/τρ = 1/τSAE is valid resembles the situation in LiTiS2 studied earlier.[42] Here, both methods probe one and the same diffusion process in Li4SnS4 that is characterized by Ea = 0.55 eV and 1/τ0 = 3 × 1013 s–1. Prefactors in this order of magnitude agree with phonon frequencies typically expected for solids.

Importantly, for Li4GeS4 the 1/τSAE Arrhenius line is shifted by almost 1 order of magnitude toward lower rates. This shift reveals that 7Li SAE NMR is sensitive to the same diffusion processes which produce the spin-lock NMR rate peak at 360 K. Compared to Li4SnS4, stimulated echo NMR tells us that Li+ diffusion is by a factor of ∼8 lower in the Ge sulfide. The same difference is seen in total conductivity of our recent impedance spectroscopy study focusing on the Li4Ge1–SnS4 (0 ≤ x ≤ 1) series.[36]

Finally, 7Li SAE NMR reveals the difference in the so-called final amplitudes S∞ of the S2 curves shown in Figure b. At sufficiently long evolution time tp, and strictly speaking only for pure spin-alignment order, S∞ is proportional to the inverse number of electrically inequivalent sites participating in Li+ diffusion.[59] Dipolar coupling can, however, decrease this value.[45,60] It is interesting to note that the Sn samples show a relatively high amplitude of S∞ = 0.1, while that of the Ge-containing samples is reduced by a factor of 2. The difference mirrors the quadrupole coupling strength shown in Figure b. Noteworthy, while S∞ is affected by the choice of tp, it turned out that the decay rate 1/τSAE is largely unaffected by tp if varied from 15 to 100 μs (see Figure S3). In general, tp, especially in 6Li SAE NMR, controls the sensitivity of an SAE NMR experiment with respect to the loss of phase coherence. Here, the introduction of dipolar couplings introduced at longer preparation times has no effect on 1/τSAE [≠ f(tp)]; thus, we assume the ions sense sufficiently large temporal fluctuations in electric field gradients during the diffusion process so that all Li+ jumps relevant for SAE NMR decay are sampled at short tp.

Comparison with Conductivity Data and Structural Features

From a structural point of view, the four samples differ in Li+ population of the available sites in Li4MCh4. Quite recently, we showed that the Li polyhedral volume LiCh greatly affects both the Li+ distribution and Li ion transport properties as probed by impedance spectroscopy performed on the same samples as studied here.[35,36] Our recent measurements point to an optimal volume for Li+ diffusivity that is located between 13.0 and 13.5 Å.[35] Time-domain 7Li NMR spectroscopy underpins this result as can be seen in Figure a,b, which shows both the change of Ea,SAE (and Ea,ρ) and that of 1/τ0 as a function of Li polyhedral volume for the four compounds studied. Disregarding the results for the relaxation peak seen at 360 K (Li4GeS4), the parameters pass through minima located near 13 Å, so close to the LiCh volume of Li4SnS4. Figure a again illustrates the perfect agreement found when activation energies from spin-lock NMR are compared with those from two-time stimulated echo NMR.

Figure 4

(a) Change of activation energies probed by both 7Li spin-lock NMR (Ea,ρ) and 7Li SAE NMR (Ea,SAE) as a function of the LiCh volume in Li4MCh4 (M = Sn, Ge; Ch = S, Se). The value of the second rate peak of Li4GeS4 is also included in parentheses. (b) Variation of the pre-exponential factor of 7Li NMR spin–lattice relaxation rate measurements performed at a (technical) locking frequency of 20 kHz.

Conclusions

Ternary Li4MCh4 served as a model system to study Li+ ion dynamics by three time-domain 7Li NMR methods including spin–lattice relaxation in both the rotating and laboratory frame of reference and 7Li two-time SAE NMR. For the four samples investigated we recognized that laboratory spin–lattice relaxation rate is sensitive to fast (localized) jump process governed by relatively low activation energies (0.17–0.38 eV). Though governed by low coupling constants and thus being influenced by nondiffusive background relaxation, diffusion-controlled peaks 1/T1(1/T) appear at temperatures as low as 290 K (Li4GeSe4). These localized motions are to be characterized by jump rates in the order of 109 s–1.

Ionic diffusion on a longer length scale was, however, probed by spin-lock 7Li NMR performed at a locking frequency of ω1/2π = 20 kHz. Complete diffusion-induced rate peaks 1/T1ρ(1/T) were observed for all samples with peak maxima appearing at 320 and 328 K, respectively. In addition, for Li4GeS4 another rate peak at 360 K is seen indicating a complex Li+ behavior with at least two dynamically distinct processes occurring in parallel. The corresponding activation energies of the almost symmetric BPP-type peaks range from 0.53 eV (Li4SnS4) to 0.69 eV (Li4SnSe4), partly exceeding corresponding values from impedance spectroscopy. However, mixing-time-dependent 7Li SAE NMR corroborates the activation energies deduced from spin-lock NMR. Moreover, for Li4SnS4 the absolute motional correlation rates 1/τρ and 1/τSAE perfectly match and show that stimulated echo NMR provides a direct, model-independent access to the Li+ jump rates.

Taken together, combining these results with those from 7Li NMR line shape measurements, for each of the four compounds investigated, we probed a diffusion process that is not overwhelmingly governed by correlation effects or large distributions of motional correlation rates. The exceptionally high activation energy Ea of Li4SnSe4 (Ea,ρ = 0.69 eV ≈ Ea,SAE = 0.72 eV) in conjunction with almost exponentially decaying two-time SAE NMR decay curves (stretching parameter of n = 0.92) points to virtually uncorrelated Li+ ionic motions being at the same time subjected to a very high attempt frequency (∝ 1/τ0) almost reaching 1016 s–1. In the other samples, with Li4GeSe4 taking an intermediate position, we suspect correlation effects or concerted motions coming gently into play, which reduce both Ea and 1/τ0.