Abstract
Fundamental research on lithium ion dynamics in solids is important to develop functional materials for, e.g. sensors or energy storage systems. In many cases a comprehensive understanding is only possible if experimental data are compared with predictions from diffusion models. Nuclear magnetic resonance (NMR), besides other techniques such as mass tracer or conductivity measurements, is known as a versatile tool to investigate ion dynamics. Among the various time-domain NMR techniques, NMR relaxometry, in particular, serves not only to measure diffusion parameters, such as jump rates and activation energies, it is also useful to collect information on the dimensionality of the underlying diffusion process. The latter is possible if both the temperature and, even more important, the frequency dependence of the diffusion-induced relaxation rates of actually polycrystalline materials is analyzed. Here we present some recent systematic relaxometry case studies using model systems that exhibit spatially restricted Li ion diffusion. Whenever possible we compare our results with data from other techniques as well as current relaxation models developed for 2D and 1D diffusion. As an example, 2D ionic motion has been verified for the hexagonal form of LiBH4; in the high-temperature limit the diffusion-induced 7Li NMR spin-lattice relaxation rates follow a logarithmic frequency dependence as is expected from models introduced for 2D diffusion. A similar behavior has been found for LixNbS2. In Li12Si7 a quasi-1D diffusion process seems to be present that is characterized by a square root frequency dependence and a temperature behavior of the 7Li NMR spin-lattice relaxation rates as predicted. Most likely, parts of the Li ions diffuse along the Si5 rings that form chains in the Zintl phase.
1 Introduction
Fundamental research on lithium diffusion in solids using nuclear magnetic resonance (NMR) has been carried out since the 1950s. Very first examples include the study of Li self-diffusion in lithium metal by temperature-dependent NMR spin-lattice relaxation (SLR) [1], [2]. Until today quite many NMR methods have been proven as versatile tools to study Li ion dynamics in numerous compounds such as crystalline inorganic materials, glasses, polymers and so forth [3], [4]. NMR offers a powerful collection of techniques with which dynamics in solids and liquids can be probed over a large times scale. The majority of the studies published over the last decades are concerned with the measurements of jump rates and activation energies as well as the elucidation of geometric aspects of translational and rotational dynamics. Time-domain SLR NMR is the method of choice to study Li ion self-diffusion processes if information on jump rates and activation energies are needed [5], [6], [7], [8], [9]. With NMR relaxometry it is possible to probe both local hopping processes as well as long-range ion transport [7], [10].
Most interestingly, in favorite cases NMR is also able to distinguish between diffusion processes characterized by their dimensionality [11], [12]. For that purpose powder samples are sufficient and there is no need to investigate single crystals. This information is included in both the temperature and frequency dependence of the corresponding diffusion-induced 6Li and 7Li NMR SLR rates. As yet, only a few studies took up the challenge to use NMR relaxometry to describe low-dimensional translational Li ion dynamics in crystalline materials [10], [12], [13]. If ionic jump diffusion is utterly restricted to two dimensions or even to only one dimension, the diffusion-induced relaxation rates are expected to show a characteristic temperature and frequency dependence being an immediate consequence of the constraints of the lattice.
The deviations expected for 2D and 1D are referred to the so-called standard behavior that applies to isotropic, i.e. 3D, uncorrelated jump diffusion. As early as 1948 Bloembergen, Purcell and Pound (BPP) described this type of motion with a single exponential correlation function G(t), which characterizes the temporal evolution of the internal fluctuating dipolar magnetic fields the diffusing ions are subjected to [14]. The Fourier transform of G(t) leads to the spectral density function J(ω); the spin-lattice relaxation rate T1−1, being the primary observable in NMR relaxometry if carried out in the laboratory frame of reference, is directly proportional to J(ω0). ω0 denotes the angular resonance frequency, the Larmor frequency, of the nucleus under investigation. Assuming Arrhenius behavior for the correlation rate τc−1 characterizing the decay of G(t), the ideal BPP-type response leads to a symmetric relaxation rate peak if T1−1 is plotted on a logarithmic scale vs. the inverse temperature. While at the peak maximum the correlation rate can be estimated according to ω0τc≈1, usually τc is very similar to the mean jump rate τ, the flanks contain information on (i) motional correlation effects and (ii) the dimensionality of the diffusion process. Correlations of successive jumps, e.g. due to Coulomb interactions or structural disorder [15], cause deviations from the quadratic frequency dependence of T1−1 in the low-temperature regime, i.e. in the regime characterized by ω0τc≫1. If, on the other hand, we have to deal with 2D or 1D diffusion a characteristic asymmetry of the SLR NMR rate peak is predicted by the (semi-empirical) relaxation models introduced by Richards (2D) [16] and Sholl (1D) [17]. Most importantly, in the high temperature limit of the peak (ω0τc1) the rates should reveal characteristic dependencies on ω0. This behavior is in stark contrast to 3D diffusion for which no such dependence should show up. While for 1D diffusion the relaxation models predict a square root dependence, a logarithmic one should govern jump diffusion being confined to two dimensions; see the summary in Table 1, adapted from Ref. [5]. The latter case is expected to be met in, e.g. layered materials with the Li ions located between the layers. In principle, the same holds true for spin-lock SLR NMR performed in the rotating frame of reference (spin-locking angular frequency ω1ω0) whereby the diffusion-induced peak is shifted to lower temperatures. Up to now, only few examples have, however, been presented that clearly support these predictions by theory [11], [18].
ω0τc≪1 | |
---|---|
3D | ∝ τc |
2D | ∝ τcln(1/ω0τc) |
1D | ∝ τc(ω0τc)−0.5 |
Worth noting, the frequency dependence of the high-T flank of the T1−1 rates is often difficult to access experimentally. Then, it can be supported by field-cycling (FC) NMR measurements of SLR rates. FC NMR enables direct access to T1−1(ω) over a wide frequency range, see Ref. [19], and may help us to further characterize low-dimensional jump diffusion.
In the following we will present a short overview of recently studied compounds in our labs. Whereas earlier investigations put emphasis on 2D diffusion in LixTiS2 and the quasi 1D diffusion in β-eucryptite (LiAlSiO4), for example see Refs. [7], [11], [20], [21], [22], [23], [24], [25], [26], [27], the present collection comprises, on the one hand, layer structured materials such as hexagonal LiBH4 [12], [28], LixNbS2 [29], [30], Li0.17SnS2 [31], LiC6 [32] and, in a companion article, Li2Ti3O7 (as well as isostructured Na2Ti3O7) [13]. On the other hand some silicides, sulfides and oxides with channels are treated. While for Li0.7Nb3S4, LixNa2−x Ti6O13 [33], LiBi3S5 [34] and ramsdellite-type Li2Ti3O7 [35], [36], [37] the ions are expected to move inside the channels, the stacked Si5 rings in Li12Si7 [38], [39], [40] guide the ions outside the channels to follow quasi 1D diffusion.
2 Li-ion conductors with two-dimensional diffusion pathways
2.1 LiBH4
The hexagonal modification of lithium boron hydride [41], [42], [43] turned out to be a suitable model system to study 2D ionic diffusion [12], [28]. Below the phase transition temperature, Tp, of 380 K lithium boron hydride crystallizes with orthorhombic symmetry (Pnma). The crystal structure of the orthorhombic modification allows for 3D diffusion. Translational Li ion diffusion is, however, much slower than the rotational dynamics of the BH4 anion unit. Thus, the 6Li and 7Li NMR SLR rates are completely governed by the dynamics of their surroundings [12]. This great difference in Li ion diffusivity is best seen if the corresponding 7Li NMR and 6Li NMR spectra are regarded [12], [28]. Whereas in orthorhombic LiBH4, because of poor Li exchange, the lines are dipolarly broadened, sharp NMR lines show up in the layered variant. The lines meet all the expectations of a system with fast diffusing spin-3/2 quadrupolar nuclei that are exposed to a mean electric field gradient. The jump in Li ion diffusivity, occurring when going from orthorhombic to hexagonal LiBH4 (P63mc or P63/mmc, see Ref. [44]), is also seen when dc conductivity, σdc, values are considered. Just above the phase transformation the ionic conductivity rapidly rises by two orders of magnitude, see Figure 1 and the conductivity isotherms in (b), [43], [45], [46].
To decide whether Li diffusion in layer-structured, hexagonal LiBH4 is indeed of two-dimensional nature, we recorded diffusion-induced SLR NMR rates. We could gather the high-temperature flank that contains the information on the dimensionality of the diffusion process (see Table 1). Since the relaxation models useful to differentiate between 2D and 3D diffusion were developed for dipolar interactions, we focused on 6Li NMR. Although the 6Li nucleus is a quadrupolar one with a spin-quantum number of I=1, its quadrupole moment is much lower, by a factor of 50, than that of 7Li (I=3/2). Therefore, it can be regarded as a quasi spin-1/2 nucleus that is mainly subjected to dipole-dipole interactions. In Figure 2a the 6Li NMR spin-lattice relaxation rate peaks are shown that illustrate low-dimensional Li ion diffusion in hexagonal LiBH4. As expected, the rate peaks shift toward higher T the higher the Larmor frequency chosen to record the rates. Most importantly, besides an almost quadratic frequency dependence on the low-T side, the rates of the high-T flank, where ω0τ≫1 holds, do not coincide as it would be expected for 3D diffusion. Instead, both rate peaks could be satisfactorily parameterized with the semi-empirical model introduced by Richards [16], [47]. This model has been developed for 2D diffusion. It takes into account a smaller slope of the high-T flank. For comparison, uncorrelated 3D motion [5], [14] would lead to symmetric rate peaks (see the dashed line in Figure 2a). The model of Richards predicts a logarithmic frequency dependence of the diffusion-induced SLR NMR rates in the limit ω0τ≫1. In order to prove this prediction we recorded the rate 1/T1 as a function of frequency at 483 K, i.e. on the high-T side of the diffusion-induced rate peaks. The result is shown in Figure 2b, the rate indeed follows a linear behavior if 1/T1 is plotted vs. ln ω0. This behavior definitely proves the validity of the model of Richards and the 2D nature of Li ion diffusion in layer-structured LiBH4.
Recently, the 2D diffusion process was also confirmed by theory, see, for example the ab initio molecular dynamics study performed by Aeberhard et al. [48] stating that Li diffusion in LiBH4 is effectively two dimensional in nature. Density functional theory calculations and quasi-elastic neutron scattering experiments of Myrdal et al. [49] also showed that the most favorable mechanism for Li+ conduction takes place in the hexagonal plane; their calculations revealed that jump processes with a component in the z-direction have to be characterized by higher energy barriers, thus supporting the idea of a two-dimensional transport process.
As a last remark, we would like to draw the reader’s attention once again to the fact that we found indications for uncorrelated Li diffusion [28]. The solid lines in Figure 2a include a quadratic frequency dependence in the low-T regime. This proportionality, 1/T1∝ω0−2, is predicted by the BPP model that is, as mentioned above, based on an exponential motional correlation function G(t). The activation energy (0.57 eV) extracted from 6Li SLR NMR measurements carried out at 58.9 MHz agrees well with that obtained from conductivity measurements (0.55 eV). Therefore, we conclude that the ions are exposed to a rather regular potential landscape with no differences between short-range and long-range ion dynamics. The corresponding conductivity isotherms of hexagonal LiBH4 do not reveal a pronounced dispersive region which would be indicative of correlated jump processes [50]. Instead the conductivity plateau directly discharges into the regime where the real part of the complex conductivity is governed by lattice vibrations, i.e. characterized by σ′∝υ2, see Figure 1b.
2.2 LixNbS2
Layer-structured transition metal chalcogenides are ideally suited to host foreign atoms such as Li+ or Ag+ [25]. Self-diffusion of these guest atoms is anticipated to mainly occur within the van-der-Waals gap, i.e. between the transition metal oxide or sulfide layers. For instance, earlier NMR relaxation studies revealed 2D Li ion diffusion in the layer-structured 1T-modification of LixTiS2 for x considerably lower than one [11], [26]. In Li0.7TiS2 a single diffusion process was probed covering a dynamic range of almost 10 orders of magnitude [26], [27]. In the 3R as well as the 2H phase of LixNbS2 the Li ions have also access to intralayer diffusion; its low dimensionality could be revealed via frequency-dependent 7Li NMR spin-lock relaxation measurements [29], [30]. As (first-order) quadrupole interactions in LixNbS2 turned out to be relatively small, the 7Li nucleus served as a suitable probe for our analysis (Figure 3).
In contrast to LiBH4 for 3R-Li0.7NbS2 much higher temperatures (>500 K) are, however, needed to record the high-T flank of the diffusion-induced rate peaks 1/T1(1/T). Therefore, we took advantage of spin-lock relaxometry that is usually carried out at locking frequencies ω1/2π in the kHz range [29]. With ω1/2π ranging from 5 to 40 kHz the peaks 1/T1ρ(1/T) show up between 285 K and 330 K clearly shifting toward higher T with increasing locking frequency. In Figure 3a and b the frequency dependence of the 7Li SLR NMR rate in the limits ω1τ1 (358 K) and ω1τ≫1 (270 K) are shown. In the low-T range we observed a slight deviation from uncorrelated motion that is expressed by the exponent β being, for correlated motion, smaller than two. Most importantly, for ω1τ1 the spin-lock SLR NMR rates linearly depend on ln ω1 pointing to 2D Li ion diffusion. As for LiBH4 the absolute change of the rates with increasing ω1 turned out to be measurable, but was low. This is even more pronounced for 2H-LixNbS2 (see Figure 4). We assumed a wavelike 2D diffusion process, i.e. an oscillating trajectory which results if we consider the various sites within the van-der-Waals gap acting as transition states. Such transition states might be the tetrahedral sites connecting two octahedral voids by sharing common faces; a similar diffusion pathway was earlier found for LixTiS2 (x≤0.7) by means of 7Li stimulated echo NMR [7], [27].
For 2H-LixNbS2 we investigated three samples with x=0.3, 0.7 and 1.0 [30]. 7Li SLR NMR in both the laboratory and rotating frame of reference were used to quantify the effect of correlation effects on Li ion diffusion with increasing charge carrier density. If analyzing 1/T1ρ with the 2D spectral density function of Richards, we were able to clearly witness that Li ion self-diffusivity is slowed down with increasing x [30]. Activation energies Ea ranged from 0.44 eV (x=0.3) and 0.48 eV (x=0.7) to 0.64 eV (x=1.0); accordingly, the rate peaks systematically shifted to higher T the larger x. Considering, however, the activation energies from laboratory frame 1/T1 in the limit ω0τ≫1 we observed the opposite trend: At the same time, the β parameter used to describe the asymmetry of the 1/T1ρ rate peaks decreased with increasing x [β=1.8, (x=0.3), β=1.7 (x=0.7), and β=1.4 (x=0.3)] [30]. This observation was interpreted in terms of an increasing influence of correlation effects due to enhanced Coulomb interactions for samples with a high concentration of mobile Li ions [30]. Such effects lead to lower slopes in the low-T limit than expected for ion jump diffusion. Hence, an Ea that decreases with increasing x turned out to be measure for correlation effects rather than a parameter to describe energy barriers on the angstrom length scale. Of course, to a small extent, the widening of the van-der-Waals gap with increasing Li content might reduce the local barriers the ions have to surmount. In our case, we have, however, to take into account two facts. First, we deal with a rather regular potential landscape and second we saw that long-range ion diffusivity decreases with increasing x. Therefore, we could not find convincing arguments that would support the idea of enhanced local motions the higher the Li content, not even if we consider interlayer diffusion processes via defects or interstitial sites.
2.3 Li0.17SnS2
Apart from LiBH4, LixTiS2 and LixNbS2 the Sn-analogue Li0.17SnS2 was used as another model substance to probe low-dimensional diffusion [31]. In contrast to LixTiS2 in SnS2 the intra- and interlayer octahedral and tetrahedral sites may act as hosts for Li ions, see Figure 5a. According to Pietrass et al. [51] the amount of Li ions inserted needs to be relatively low to avoid the occupancy of voids in the SnS2 layers. Thus, we focused on a sample with x=0.17; at this low Li content also the influence of staging effects can be ruled out [52]. As opposed to LixNbS2 and LixTiS2 for a non-annealed sample we observed two overlapping relaxation rate peaks 1/T1ρ whose amplitudes change upon annealing the sample (see Figure 5b). We assume that peak 1 represents those ions that form clusters near the surface-influenced regions of the SnS2 crystallites. In such regions the ions may have access to short-ranged diffusion pathways. Upon annealing the ions became more uniformly distributed among the available octahedral sites between the SnS2 sheets. As a consequence, peak 1 drastically loses in intensity. The final NMR rates can either be parameterized with two rate peaks or, roughly speaking, with only one rate peak that agrees with 2D diffusion (see below). Due to the complex ion dynamics observed in Li0.17SnS2 and the heat sensitivity of the samples we passed further frequency-dependent NMR measurements up. The 2D nature of Li ion diffusion in Li0.17SnS2, however, indirectly showed up when analyzing the NMR response before annealing.
Regarding the situation before heat treatment, peak 2 was approximated with the expression of Richards for 2D diffusion [16], [47]. For peak 2 we used either a BPP-type (3D) or 2D ansatz [31]. The overall fit is shown in Figure 5b as a thick solid line. The underlying two fits of peaks 1 and 2 are given by dashed lines. If peak 2 is approximated with the spectral density for 2D diffusion it automatically reflects the NMR response that is obtained after the annealing step. Note that in Figure 5b the data points for the annealed sample are shown for comparison only.
2.4 LiC6
Graphite is one of the best known 2D energy materials; it represents an excellent anode material in Li ion batteries. Li ions can be reversibly inserted and de-inserted without any severe structural change of the host. The stage-1 compound LiC6 is a gold-colored substance. It consists of commensurate layers of carbon and lithium; Li resides in the central site below and above each third hexagon formed by C atoms (see Figure 6). The stacking sequence is AαAα with a periodicity of 3.71 Å; A denotes the graphite basal plane and α represents the intercalate layer. The ions have access to several hopping processes that have been the subject of some calculations on ion dynamics in graphite intercalation compounds. The host structure allows for in-plane jump diffusion, see Figure 6. In earlier studies it has been argued that diffusion perpendicular to the c-axis cannot be ruled out completely [53]. Vacant C sites will certainly promote interlayer Li diffusion.
Although ion dynamics in this mixed-conducting compound has been investigated in the past via quasi-elastic neutron scattering (QENS) and NMR [54], [55], [56], for example we re-investigated ion self-diffusion by means of 7Li spin-spin and spin-lock NMR relaxometry [32]. Whereas the laboratory 7Li SLR NMR rates were strongly affected by the Li+-electron interactions in LiC6 (see Figure 7a), the rates in the rotating frame of reference as well as the corresponding spin-spin relaxation rates were useful to extract diffusion parameters.
The rates 1/T1ρ pass through a diffusion-induced rate peak which is located at 314 K. It can be best represented with a Lorentzian-shaped BPP-type spectral density function. The asymmetry parameter turned out to be ca. 1.7. For Ea we obtained 0.57 eV; the activation energy of the low-T side is only 0.42 eV indicating correlated motion [32]. This value is comparable with that from 7Li NMR spin-spin relaxation (0.53 eV) and agrees well with the result from Figure 7b. In Figure 7b Li jump rates from the various NMR methods, including previous results from 8Li β-NMR on oriented LiC6 crystals [54], [55], and QENS [56] are shown in a joint Arrhenius plot. The Arrhenius line drawn is characterized by 0.55 eV; this activation energy agrees well with results from theory for both the in-plane vacancy and interstitial Li migration mechanism [57], [58]. Residence times in the order of 1 μs are obtained at ca. 350 K. At temperatures below the melting point of water we have to deal with residence times in the order of 104 s−1. Such information has enormous practical importance. At low T the slow kinetics and the low Li+ diffusion coefficients in graphite battery anodes promote Li plating rather than Li insertion. The deposition of metallic Li on the graphite surface is highly problematic in terms of safety and life cycle of these storage systems. 7Li-NMR and impedance studies dealing with practical aspects of graphite anode materials are, e.g. presented in Refs. [59], [60].
3 Li compounds proffering one-dimensional diffusion
3.1 Li12Si7
In polycrystalline Li12Si7, being one of the Li-rich Zintl phases in the binary Li-Si system, strong evidence for one-dimensional Li motion has been reported [10], [38], [39], [40]. This 1D Li+ motion can be linked to channel-like columns formed by the stacked planar silicon rings which guide the Li ions in the close proximity of the channels. The Li ions sandwiched between the rings are much less mobile as has been verified by 6Li 2D exchange NMR [39]. These ions can be identified by their characteristic isotropic chemical shift of−17.2 ppm in 6Li magic angle spinning (MAS) NMR. The, at first glance, unexpected shift to a higher magnetic field has been explained by the strong shielding effects caused by an aromatic ring current of the Si5 rings [38].
In Figure 8a, the diffusion-induced 7Li NMR SLR rates 1/T1 and 1/T1ρ are plotted versus the inverse temperature for different Larmor frequencies ω0 and spin-lock angular frequencies ω1, respectively. The spin-lock SLR rates clearly reveal at least three maxima, one being already visible at very low temperatures. In line with relaxation models for 1D diffusion (cf. Table 1) the activation energy of the high-T flank is given by EA/2=0.09 eV, that is, half the value of the activation energy obtained in the regime ω0τ≫1. Together with the frequency dependence observed this asymmetry represents a strong indication of 1D diffusion. This quasi 1D Li motion is also seen in our relaxometry measurements performed in the laboratory frame of reference, see Figure 8a; the corresponding diffusion-induced 7Li SLR NMR rate peaks also reveal a dispersive behavior in the limit ω0τ1. Taking 1/T1(ω0) and 1/T1ρ(ω1) together, the data fulfill the square root frequency dependence (cf. Table 1) over three orders of magnitude [40]. An analogous result for 2D Li diffusion was early found in the case of the layered model system LixTiS2 [5], [11].
The other spin-lock SLR rate peaks showing up at higher T have been assigned to two different 3D Li diffusion processes being consecutively activated with increasing temperature. As is diagnostic for 3D diffusion they are symmetric in shape and show no frequency dependence on the respective high-T flanks. The corresponding activation energies turned out to be 0.32 eV and 0.55 eV.
In Figure 8b the jump rates deduced from the various NMR rate peak maxima are compared with results from other NMR investigations; these techniques include the analysis of NMR MAS spectra, NMR exchange spectroscopy, the analysis of 7Li NMR quadrupolar satellites and 7Li NMR line widths. Moreover, Li diffusion coefficients from static (SFG) and pulsed field-gradient (PFG) NMR as well as electrochemical investigations (galvanostatic intermittent titration technique, GITT) are shown. The diffusion coefficients measured by 7Li SFG NMR and PFG NMR were converted to jump rates using the Einstein-Smoluchowski equation. FG NMR shows that the 1D Li motion observed with 7Li NMR relaxometry is a fast, long-range diffusion process. In a recent study [39] it has been shown that in Li12Si7 9 of 13 Li positions contribute to this jump process. The other two diffusion processes, which are 3D in nature, correspond to Li ions being stronger bound to the Si lattice and having medium and low diffusivity, respectively [38]. Our experimental findings were recently corroborated in a dedicated atomistic study using density functional theory by Shi et al., see Ref. [61].
3.2 Li0.7Nb3S4
In contrast to Li12Si7 with its multiple Li sites, in LixNb3S4 [62] the guest ions solely reside in the channels formed by the Nb-S units. Li motion is supposed to take place mainly within these channels. For isostructural Li0.7Nb3Se4, Kriger et al. [63] found indications for inter-channel and intra-channel Li hopping. We employed 7Li NMR SLR to study the effects of dimensionality on Li motion in Li0.7Nb3S4.
In Figure 9a, the 7Li NMR SLR 1/T1 and spin-lock SLR rates 1/T1ρ, which were recorded at the frequencies indicated, are plotted versus the inverse temperature. For the laboratory-frame SLR rates 1/T1, no maximum could be detected; with the setup used only the low-temperature flank was reached. To obtain the diffusion-induced SLR rates, the total SLR rates were corrected for the non-diffusive background rates whose temperature dependence was approximated by a power law T−α with α≈2. The slope of the low-T flank of the purely diffusion-induced SLR rate yielded an activation energy of EA=0.25(2) eV, which we attribute to short-range Li-ion movements. Most likely, the slight change in slope at 440 K indicates a change in diffusion mechanism.
Information on slower Li diffusion processes in Li0.7Nb3Se4 were obtained by spin-lock 7Li SLR NMR relaxometry for frequencies of 10 kHz, 15 kHz, and 20 kHz, respectively. The corresponding rates were also corrected by the power law for any non-diffusive background effects; α ranged from 1 to 2.5. For all three frequencies, relaxation rate maxima were observed. The continuous lines show fits with a BPP-type relaxation model used to calculate the position of the peaks and to estimate the activation energy in the regime ω1τ≫1. Here, the fits yielded EA=0.65(6) eV; such a relatively large activation energy might indeed be attributed to channel-like Li diffusion. Note that activation energies deduced in the limit ω0τ≫1 can be influenced by correlation effects reducing the slope of the rate peak in this limit.
The view of slow spatially restricted Li diffusion is supported by the double-logarithmic plot of the diffusion-induced spin-lock NMR SLR rate vs. the locking frequency (533 K). The frequency dependence observed is not in contradiction to a slow 1D low-diffusion process in Li0.7Nb3S4 [see inset of Figure 9a].
Li jump rates were also determined by 7Li NMR spin-alignment echo (SAE) measurements; the corresponding decay rates are shown together with the results from spin-locking 7Li NMR in Figure 9b. If taken together an activation energy of ca. 0.9 eV is obtained clearly pointing to sluggish Li ion transport. The pre-factor of the underlying Arrhenius law is in the order of 1014 s−1. This value is in the order of phonon frequencies and, thus, comparable to those of other Li ion conductors such as Li3N, Li4Ti5O12, Li7BiO6, or h-Li0.7TiS2 [27], [64], [65], [66], [67], [68].
3.3 Mixed-alkali LixNa2-x Ti6O13
Another potential candidate to observe one-dimensional Li diffusion is Li2Ti6O13. In contrast to Li0.7Nb3S4, Li2Ti6O13 does not act as a classical intercalation material. Li insertion occurs through an ion exchange reaction using the parent compound Na2Ti6O13 [69]. The Li ion occupies the same sites as Na and the two crystallographic sites available are identical. 23Na and 6Li as well as 7Li NMR spectra show only a single line. The end members Na2Ti6O13 and Li2Ti6O13 differ in thermal stability. Whereas Na2Ti6O13 is stable at least up to 1273 K [33], Li2Ti6O13 is only stable below 873 K [70].
The 23Na and 7Li NMR SLR rates measured are shown in Figure 10 covering a temperature range from 295 to 870 K. 23Na NMR SLR rates were recorded under MAS conditions using a spinning rate of 3 kHz. Any influence of spinning on the SLR rates was excluded; in general, such influence has been found to be low up to 5 kHz spinning frequency for 1H NMR SLR [72]. The 23Na NMR SLR rates were corrected for the non-diffusive background effects using a power law Tα, as already discussed above. The exponent α=1.1(2) indicates an interaction of 23Na with paramagnetic centers or conduction band electrons. Most likely, a change in the relaxation mechanism shows up at higher temperatures, which led to a further increase of the NMR SLR rates. In addition to the interactions with paramagnetic centers and conduction band electrons being dominant at low temperatures, it might be possible that spin-phonon coupling through quadrupolar interaction induces a further contribution to non-diffusive background relaxation at the highest temperatures [73].
After taking into account any background effects, diffusion-induced 23Na NMR SLR rates were obtained. Due to the larger quadrupolar moment of 23Na compared to 7Li, SLR is mainly caused by quadrupolar interactions of the 23Na nuclei with electric field gradients at their sites. From the 23Na SLR rate peak the Na jump rate in Na2Ti6O13 was calculated to be τ−1≈1×109 s−1 at 692 K. The activation energy was estimated to be ca. 0.82 eV.
7Li NMR SLR rates were analogously corrected for the non-diffusive background. The power law exponent turned out to be α=4.2, i.e. much higher than in case of the 23Na NMR SLR rates. Presumably, spin-phonon couplings govern non-diffusive background SLR in Li2Ti6O13 already at low temperatures. Considering the SLR rate peak we end up with τ−1≈2.4×108 s−1 at 674 K. For the activation energy a value of 0.65 eV could be estimated from the low-T flank. Future studies should include frequency-dependent measurements to collect information on the dimensionality of the slow diffusion process seen by both 23Na and 7Li NMR.
Nevertheless, first hints to low-dimensional ion motion might already be obtained from the local ionic movement in mixed Li/Na ion conductors LixNa2−x Ti6O13. Static 7Li NMR spectra were recorded at different temperatures for Li0.33Na1.67Ti6O13, LiNaTi6O13, and Li2Ti6O13. These spectra showed only the central transition and with increasing temperature narrowing of the line width sets in [see inset of Figure 10b]. Line narrowing is caused by the averaging of homonuclear and heteronuclear dipole-dipole interaction. Motional narrowing slightly depends on x. With increasing Li content, particularly if we consider the samples LiNaTi6O13 and Li2Ti6O13, a slight increase in dipolar coupling is observed which can be explained by weaker homonuclear couplings, see van Vleck’s formula [74] which was introduced to relate the spin-spin distances with the second moment of the NMR line. For LiNaTi6O13 almost the complete motional narrowing curve is seen, the residual line width in the range of extreme narrowing amounts to ca. 700 Hz. The same value, although the curve is only partly detectable with our setup, is assumed for the Li-rich compound Li2Ti6O13. For Li2Ti6O13 the onset of line narrowing is slightly shifted toward higher T as expected because of the larger rigid lattice line width. An additional shift by ca. 50 K, if compared to the situation for LiNaTi6O13, most likely also reflects a somewhat slower Li self-diffusion process in Li2Ti6O13. For Li0.33Na1.67Ti6O13 motional narrowing remains incomplete until ca. 500 K. The difference in line widths is only 300 Hz. Such a small difference is known from hydrogen diffusion in hydrides, where initial local motion or rotation leads only to a small reduction, while at higher T line narrowing continues due to several jumps over longer distances [41], [42]. As lithium is diluted in Li0.33Na1.67Ti6O13 the partial narrowing seen might be attributed to local rather than long-range motion; Li mobility in Na-rich samples seems to be restricted because of the larger sodium ions.
Activation energies were estimated from a fit of the motional narrowing data using the empirical expression by Hendrickson and Bray [71]. The activation energies varied by about 0.1 eV for the different Li/Na ratios and range from 0.7 to 0.9 eV. A similar trend was observed by 7Li NMR SLR.
In general, the LixNa2−x Ti6O13 revealed relatively low rigid-lattice line widths. This fact and the observation of partial line narrowing for diluted Li0.33Na1.67Ti6O13 shows that the Li ions interact with only few neighbored spin-carrying ions as it would be the case inside a channel. Hence, most likely Li and Na diffusion is indeed restricted to the space inside the channels of the titanate.
3.4 LiBi3S5
LiBi3S5 also crystallizes with a channel-like structure and the Li ions residing inside the channels have, thus, access to a spatially rather restricted diffusion pathway (see Figure 11b). We followed a solid-state route to prepare phase-pure gray-black LiBi3S5 powder starting from LiBiS2 and Bi2S3 [34]. Neutron diffractograms and lithium NMR spectra reveal its crystal structure to be a cation-disordered variety of the AgBi3S5 type (synthetic pavonite; monoclinic, C2/m).
Topological analyses and 7Li NMR relaxometry suggest that the lithium ions indeed diffuse along the channels in b direction involving tetrahedral voids. Figure 11 shows the temperature dependence of the 7Li NMR 1/T1 and 1/T1ρ rates of LiBi3S5. In the case of 1/T1 above 340 K the diffusion-induced flank of the corresponding rate peak shows up; analyzing the flank yields an activation energy of 0.43 eV. At low temperatures the rates depend only weakly on T. The same features are essentially seen for 1/T1ρ. Above ambient temperature the rates already increase most likely because of some localized motional processes (regime II.). At higher T we observed the flank of the 1/T1ρ(1/T) peak that yields 0.66 eV (regime I.). Although the rate 1/T1ρ runs through different motional regimes the shape of the underlying transversal transients (see Figure 11c) does not change much. The transients can be best parametrized with stretched exponentials; the stretching factor γ takes values from approximately 0.6 to 0.7. The lowest values are found for the transition regime, i.e. when going from regime II. to I., see Figure 11a. The difference in activation energies (0.43 eV vs. 0.66 eV) can be explained by correlation effects. Obviously, compared to 1/T1ρ, they have a larger impact on the 1/T1 rates being more sensitive to short-range Li ion diffusion rather than to long-range ion transport. For spatially confined ion migration such as 1D diffusion we indeed expect correlated motion and rather large activation energies.
The result obtained by NMR relaxometry is in agreement with the motional narrowing curve shown in the inset of Figure 11a. Although the rigid-lattice line width is rather low, viz. only slightly larger than 3 kHz, motional averaging of dipolar interactions extends over a rather wide temperature range. Even at 600 K the NMR line width has not reached the extreme narrowing regime. Thus, there is no doubt that Li ion diffusion in mixed-conducting LiBi3S5 is rather poor. From a structural point of view we can explain this finding because of cation disorder. Our structural analyses [34] has shown that immobile Bi3+ ions may easily block the Li diffusion pathways, thus hindering ion migration along the channels of the sulfide. In such a case, Li ion diffusion is expected to be largely mediated by defects in the various sub-lattices enabling the ions to diffuse via non-regular lattice sites.
3.5 Ramsdellite-type Li2Ti3O7
Contrary to monoclinic Li2Ti3O7 having a layered structure, which we also examined with respect to low-dimensional Li diffusion [13], orthorhombic ramsdellite-type Li2Ti3O7 is a metastable titanate with relatively wide channels providing several inequivalent sites for the Li ions. Inside the channels the ions have access to various sites offering short-range and long-range ion diffusion. Whether Li ions do also occupy sites in the titanate framework is still under debate [36], [37], [75].
Our preliminary spin-lock 7Li NMR SLR measurements are shown in Figure 12a. Above 200 K the diffusion-induced rate peaks show up. Although the peaks were measured at three different locking frequencies, we do not find significant shifts of the peak position with increasing spin-lock frequency. In contrast to the classical expectation for 1D diffusion, a significant frequency dependence in the high-T limit is absent. Moreover, also in the limit ω0τ≫1 only a shallow frequency dependence of the rate on locking frequency is seen. The solid line represents a fit taking into account the limiting behavior for channel-like diffusion, analyzing the rate peaks yields an activation energy of 0.43 eV. The arrows drawn in Figure 12a indicate the positions of the peaks on the inverse temperature scale.
The slight frequency dependence of 1/T1ρ in the limit ω0τ≫1 might be attributed to the influence of strong local fields at the Li sites. Field-cycling (FC) NMR data carried out at 293 K [76] confirmed the results from 1/T1ρ relaxometry; from 103 Hz to 5×104 Hz the SLR rates 1/T1 are almost constant (see Figure 12b). The rate 1/T1 starts to depend on frequency for values above 100 kHz. The diffusion process seen via both FC NMR and spin-locking NMR might represent local Li ion exchange within the channels; this dynamic process would be in line with the structural model proposed by Orera et al. [75] based on Li MAS NMR assuming the Li ions to only occupy intrachannel sites in Li2Ti3O7. This assumption is corroborated by a recent high-temperature neutron diffraction study showing that the Li ions exclusively occupy interstitial, i.e. intrachannel, positions in ramsdellite-like Li2Ti3O7 rather than positions in the titanate framework [36].
To further elucidate the dynamic processes in Li2Ti3O7, 6Li SAE NMR measurements [7], [64], [77] were carried out as a function of both mixing time (tm) and preparation time, tp (see Figure 12c and d). The pulse sequence introduced by Jeener and Broeckaert [78] was used to measure the two-time single-spin correlation functions S2. The curves S2(tp=const., tm), obtained by plotting the echo intensity of the stimulated echo as a function of tm, reveal a two-step decay (Figure 12d). A sum of two stretched exponentials was used to analyze the data. Presumably, the first decay step, which is characterized by the rate constant 1/τ1 represents quadrupolar spin-spin relaxation and agrees well with the independently measured spin-spin-relaxation rate T2, which is controlled by both dipolar as well as quadrupolar fluctuations, see Figure 12c. The second decay step is characterized by τ2. 1/τ2 starts to be thermally activated at temperatures higher than 200 K. The solid lines in Figure 12c represent fits with an Arrhenius relation yielding approx. 0.1 eV. This value is independent of the preparation time used to acquire the stimulated echoes. In general, with tp the sensitivity of the experiment is set with regard to electric-quadrupolar fluctuations sensed by the Li ions. Thus, one would expect the decay rate 1/τ2 to increase with increasing tp. Moreover, the increasing generation of dipolar contributions to the spin-alignment echo at enhanced tp might lead to an additional increase of 1/τ2. At sufficiently high T, quadrupolar interactions are averaged because of rapid Li exchange between the positions involved. As a consequence, stimulated echoes are no longer able to distinguish between the electrically inequivalent Li sites, and the second decay step of the S2 curves shift toward longer mixing times. Hence, the rate 1/τ2 passes through a maximum. For tp=100 μs the rates 1/τ2 superimpose a non-diffusive background contribution. At high temperatures the S2 decay seems to be influenced by spin-diffusion usually characterized by temperature independent rates with values in the order of 1 s−1.
The effective averaging of electric quadrupolar interactions at temperatures as low as 250 K in combination with rather low activation energies suggests that 6Li SAE NMR senses local Li jumps between the close-by sites within the channels. Assuming electric field gradients with coupling constants in the order of 104 Hz, between 285 K and 250 K the Li residence times should be in the order of the inverse coupling constants of 0.1 ms.
4 Summary
Time-domain solid-state nuclear magnetic resonance is a flexible tool to access dynamic parameters of mobile ions in crystalline as well as amorphous materials. The combination of relaxation NMR with field-gradient techniques, stimulated echo NMR or even field cycling NMR allows the investigation of translational ion dynamics over a wide dynamic range, covering in the best case several orders of magnitude. Here, we used selected model substances with quite different structural features to analyze Li and Na ion dynamic parameters to understand the interplay of both short-range and long-range hopping processes in crystalline solids. The compounds were chosen such that they proffer migration pathways with reduced dimensionality, thus offering layered structures (LiBH4, LiC6, LixMeS2 (Me=Ti, Nb, Sn; 0<x≤1) or channels (LiBi3S5, Li2Ti3O7, LixNb3S4, Li12Si7) to guide the ions.
In ideal cases, NMR relaxometry, regardless of being carried out in the laboratory or rotating frame of reference, is able to sense the dimensionality of the diffusion process. There is no need for single crystals. Analyzing the frequency dependence of the NMR spin-lattice relaxation rates serves as the ultimate tool of NMR spin-lattice relaxation to judge whether a diffusion process is 3D, 2D or 1D of nature. This analysis is only possible if the so-called high-temperature flank of the diffusion-induced rate peak is experimentally accessible. For instance, via frequency-dependent NMR SLR measurements we revealed 2D Li ion diffusion in the hexagonal modification of LiBH4 as well as the transition metal dichalcogenides, see the 2H and 3R form of LixNbS2. In particular, for lithium boron hydride we clearly showed the logarithmic frequency dependence of the relaxation rate 1/T1, which is in excellent agreement with the spin-lattice relaxation model for 2D diffusion proposed by Richards. While intrachannel diffusion in LiBi3S5 and LixNa2−x Ti6O13 turned out to be rather slow, most likely because of blocking effects, the wider channels in Li2Ti3O7 seem to ensure fast Li exchange. Fast quasi-1D diffusion has also been used to explain the occurrence of 1/T1ρ rate peaks in the Zintl phase Li12Si7, which showed up at unexpectedly low temperatures. Both the asymmetry of the peaks and the frequency dependence in the high-T limit suggest spatially restricted Li motions. Obviously, the stacked Si5 rings guide the ions along a channel-like diffusion pathway.
Currently, also other 2D and 1D ion conductors are being studied in our laboratories to evaluate relaxation models introduced to describe low-dimensional ion transport. The search for systems which clearly show the signatures of quasi 1D diffusion is particularly challenging. Answering the question how and to what extent dimensionality is able to influence ion dynamics is expected to advance our efforts in developing new functional materials with controllable transport parameters besides being a fundamental issue.
Acknowledgement
We appreciate financial support by the Deutsche Forschungsgemeinschaft, grants no. DFG FOR 1277 molife HE1574/11-1, 11-2, WI3600/2-1, 2-2, and LE781/15-2. We are grateful to Prof. F. Fujara and his group at TU Darmstadt for help with the FC NMR measurements and the permission to show Fig. 12b prior to publication (cf. Ref. [76]).
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