Ion dynamics in solid electrolytes for lithium batteries

3.1 Some basics of NMR relaxometry, 3D motion in solids

To extract the R ₁ rates the change of the magnetization M = M _z , which is the sum of the magnetic moments of the spins, has to be followed as a function of delay time. Usually, we use a saturation recovery pulse sequence [98] with a comb of closely spaced π/2 pulses to prepare a defined non-equilibrium state (Fig. 3). From this state, which is given by M _z (t _d =0)=0, the amplitude of M _z increases, in the simplest case exponentially, according to

$$ M_{z}(t)/M_{0} = 1 - \exp(-t_{d}/T_{1}). $$

(1)

The rate 1/T ₁≡R ₁ depends on the fluctuations of internal fields sensed during ionic motions. They can be described by a so-called correlation function G(t). According to the model introduced by Bloembergen, Purcell and Pound (BPP) [99], developed for random jump diffusion in three dimensions, G(t) is assumed to be a single exponential

$$ G(t) = G(0) \exp (-|t|/\tau_{c}). $$

(2)

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Since ω ₀ is in the order of some MHz if external magnetic fields of several Tesla are used to measure M _z (t _d ), the corresponding residence time τ(T _max) takes values in the ns regime. τ≈1 ns roughly transforms into conductivities in the order of 10⁻³S c m ⁻¹; the corresponding diffusion-induced rate peak R _{1 diff} should desirably show up at or even below ambient temperature [18]. Replacing ω ₀ formally with a much smaller frequency in the kHz regime would shift the corresponding peak toward much lower temperatures. The measurement technique that can be used for this purpose is called spin-lock NMR where transversal spin-lattice relaxation is observed in the rotating frame of reference [102, 103, 104, 105, 106]: Immediately after a π/2-pulse the magnetization M _ρ , pointing along the (−y ^′)-axis, is locked by the magnetic field B ₁ used to generate the excitation pulse (Fig. 3). The decay of \(M_{\rho (-y^{\prime })}\) is then probed as a function of the locking pulse length t _lock.[107]. In the easiest case \(M_{\rho (-y^{\prime })}\) follows an exponential decay governed by R _{1 ρ} . Because of ω ₀≫ω ₁, spin-lock NMR is able to probe slower ionic motions. In many cases this enables access to the high-T flank of the corresponding rate peak [97]. As for R ₁, the slope of this flank describes long-range ionic motion through the crystal lattice.

In the following we will review some examples where both laboratory frame (R ₁) and rotating frame (R _{1 ρ} ) NMR relaxometry has been used to characterize ion dynamics in some of the most promising solid electrolytes. Emphasis is put on extracting Li jump rates and activation energies describing both short-range and long-range ion dynamics.

3.2 2D diffusion in lithium boron hydride: LiBH₄

Lithium boron hydride (see Fig. 4) is known as a reducing agent in organic synthesis. At room temperature the orthorhombic modification is stable that exhibits only a poor ion conductivity. While the Li ions are rather immobile on the MHz time scale, fast reorientations of the BH₄ units are seen through¹¹B,¹H NMR, and indirectly through the⁷Li spins [108, 109]. At T _hex./ortho ≈ 381 K ortho-LiBH₄ reversibly transforms into its hexagonal modification, see Fig. 4, which may be used as solid electrolyte in a lithium battery (see ref. [110, 111] and Fig. 5). Hexa-LiBH₄ shows a layered structure that facilitates ion diffusion. The associated jump in ion conductivity is well documented by Maekawa et al. [112, 113]. At approximately 393 K the ionic conductivity is in the order of 2×10⁻³S c m ⁻¹ and E _a turns out to be ca. 0.51 eV. These parameters are in excellent agreement with what has been found by⁷Li NMR relaxometry carried out at temperatures above T _hex./ortho, see Fig. 4(b).

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Above the phase transition temperature, the static⁷Li NMR line is drastically narrowed because of fast Li exchange processes that average Li-Li homonuclear dipole interactions of the 3/2-spins. This corroborates that the relaxation rate peaks seen in the hexagonal regime are due to extremely fast Li⁺ translational motions. At T _max≈450 K the jump rate τ ⁻¹ is in the order of 10⁹s⁻¹. From conductivity spectroscopy, σ≈14 mS c m ⁻¹ is obtained which is in reasonable agreement with results from NMR [113]. Worth noting, the frequency dispersion of the high-T flank of the rate peaks indicates spatially confined Li ion diffusion. Indeed,⁶Li NMR spin-lattice relaxation reveals the same characteristics which can be interpreted as 2D ionic motions [70, 114, 115] in the layered structure of LiBH₄ [116]. As the⁶Li nucleus (spin-quantum number I=1) is subject to much less dipolar and quadrupolar interactions it can be described as a quasi spin-1/2 nucleus. This circumstance is important for the meaningful use of 2D relaxation models [115] which only take into account dipolar interacting spins. Interestingly, the corresponding⁶Li NMR rate peaks indeed reveal a smaller slope in the high-T range [116] which is, in combination with the frequency dependence observed, indicative for low-dimensional diffusion and in line with the semi-empirical model for 2D diffusion introduced by Richards [115]. The low-T regime can be approximated with β=2, i.e., pointing to an exponential motional correlation function controlling the magnetic-dipolar fluctuations of the low-dimensional diffusion process.

Because of the favourable ion conductivities of hexa-LiBH₄ at elevated T we evaluated the solid electrolyte in a lithium metal cell with Li₄Ti₅ O ₁₂ (LTO) as cathode material. The insertion potential of LTO is at 1.56 V vs. metallic lithium. Via cyclic voltammetry we tested the electrochemical stability of polycrystalline LiBH₄. The material was ball-milled for 5 min in a planetary mill for better processability. Currents recorded do not exceed ±20μA; with increasing cycle no. negligible decrease in current I is seen. These observations point to a rather stable electrolyte. From cycling the system with a constant current of I=100μA we deduced a specific charge transfer resistance at the LiBH₄ | Li interface of 34 Ω cm². The overall, initial conductivity of the symmetric Li | LiBH₄ | Li cell used is 2.4 mS cm⁻¹; compared to the ionic conductivity of LiBH₄ at 393 K this suggests that the overall resistance is mainly given by that of the solid electrolyte. Of course, this might change when batteries are tested at temperatures much lower than 393 K as in the present case.

Cycling the cell with potential limitation at different C-rates results in initial capacities that significantly exceed the theoretical capacity of LTO which is 175 mAh g⁻¹. Further cycling, especially if C-rates of C/10 (148 μA c m ⁻²) are considered, reveal that a stable capacity of 125 mAh g⁻¹ can be reached without any remarkable overpotentials seen in the discharge curves; this is in agreement with recent studies by Sveinbjörnsson et al. [111]. The drastic fade in capacity occurring after the first cycles has to be looked for in electrochemical side reactions and the formation of interface layers (at the cathode side). Further fading because of increased C-rates might be traced back to kinetic reasons. Nevertheless, capacities in the order of 125 mAh g⁻¹ at C/10 mark a starting point for further research. Pure LiBH₄ can also be replaced with LiBH₄:LiI solid solutions; the incorporation of LiI stabilizes the highly conducting hexagonal modification of LiBH₄ down to lower temperatures [111, 113].

3.3 Garnet-type oxides: cation-stabilized cubic LLZO and LLZMO

The chemical robustness and high ionic, but negligible electronic, conductivities of various garnet-type oxides [19] are beneficial for the design of all-solid-state lithium batteries with metal anodes. Regarding Li₇La₃Zr₂ O ₁₂, its tetragonal modification is a moderate Li ion conductor [119]. The replacement of La and Zr ions with isovalent or aliovalent cations greatly affects, however, (i) the defect chemistry, (ii) the crystal structure parameters and (iii) also the Li content of the garnets [40, 120, 121, 122]. These variations lead to a series of oxides that may largely differ in ionic conductivities [19]. Furthermore, grain boundary structures and distinct morphologies because of the different preparation routes used influence overall ion transport, too [19].

If trivalent cations such as Al³⁺ or Ga³⁺ are used to replace the Li ions by a few percent, vacant sites in the Li sublattice are introduced. The small change in lattice constants as well as the exitance of vacant Li sites immediately causes the tetragonal form to adopt cubic symmetry [123]. The Li ions, which originally have occupied the 16f and 32g sites in tetragonal LLZO, now reside on the 96h positions which corresponds to the empty 16e site in tetra-LLZO; ions on 8a in tetra-LLZO now occupy the 24d position in the cubic polymorph [123]. In Al-stabilized cubic-LLZO the trivalent “dopants” also reside on the 24d and 96h sites [124]. Since many of these sites are vacant, facile Li ion diffusivity can be expected; indeed, Al-bearing c-LLZO belongs to the fastest oxide ion conductors known [39]. The arrangement of the Li sites is sketched in Fig. 6, see also ref. [118]. Li diffusion along the 3D 24d-96h-24d diffusion pathways (see Fig. 7(c)) is expected to largely depend on the Li content, the dopant distribution as well as the kind and number of defects, mainly vacancies, in the vicinity of the Li ions [22]. Thus, we expect a heterogeneous potential landscape with many correlated jump processes running in parallel and for which the activation energies for forward and backward jumps will also differ. The immobile trivalent dopant ions may easily block some of the fast diffusion pathways, thus clearly interacting with the mobile charge carriers.

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Through conductivity spectroscopy the average activation energy for long-range ion transport in Al-stabilized c-LLZO is given by ca. 0.35 eV [39]. Similar values have also been deduced from NMR relaxometry (see below and ref. [39]); the temperature dependence of the rates R _{1 diff} and R _{1 ρ diff}, however, clearly reveals complex length-scale dependent ion dynamics, see Fig. 7. While in the limit ω ₀ τ≫1 the flank of R _{1 diff}(1/T) is governed by only 0.12 eV, R _{1 ρ diff} data in the range ω ₁ τ≪1 points to larger (mean) barriers the ions have to surmount. Other garnets, not necessarily those with trivalent cations on 24d and 96h, have to be characterized by the same R _{1 diff} behavior [118], see Fig. 7(b).

Another almost universal feature of R _{1 ρ diff} in the case of cation-disordered LLZO is the broad relaxation rate peak that spans an extraordinary large temperature range [39, 117]. It is definitely the result of a superposition of distinct local and long-range jump processes. This is in contrast to crystal structures with similarly high ionic diffusivity but with no foreign ions on the Li sites [118]. As an example, in Fig. 7 the diffusion-induced NMR relaxation rates of polycrystalline Al-free Li_6.5La₃Zr_1.75Mo_0.25 O ₁₂ (LLZMO) are shown. In LLZMO the Li-sublattice is not influenced by blocking dopant ions and the R _{1 ρ diff} data can be approximated with two individual rate peaks. These peaks represent two almost similarly activated jump processes that differ in the pre-exponential factor of the underlying Arrhenius equation. From the peak, which is shifted toward higher temperatures, at T _max=315 K a jump rate of 4.2×10⁵s⁻¹ is obtained. This translates, using the Einstein-Smoluchowski equation, into a self-diffusion coefficient D of 1.9×10⁻¹¹cm² s⁻¹ [118].

The jump rate obtained via spin-lock NMR agrees well with that directly measured through⁷Li stimulated (spin-alignment) echo (SAE) NMR measurements [118]. With SAE NMR Li jumps between electrically inequivalent Li sites are probed. For the present case, this principle means that the garnet sites 24d and 96h are definitely involved in Li jump diffusion. Note that the SAE NMR method [38, 66, 67, 68, 69, 70, 96, 126, 127, 128, 129, 130, 131] is able to sense very low exchange rates <10⁵s⁻¹.

3.4 Phosphates with NASICON structure: Li_1.5Al_0.5Ti_1.5 PO ₄

The titanium phosphate Li_{1 + x} Al _x Ti_2−x PO ₄ with x=0.5 is one of the most promising compounds of the LATP family with ionic conductivities high enough to realize solid-state lithium batteries [65]. Its electrochemical stability against Li metal and other high-voltage cathode materials needs to be ensured by protective coatings. Via NMR we have investigated bulk ion dynamics in LATP with the composition Li_1.5Al_0.5Ti_1.5 PO ₄ [125]. In contrast to a recent NMR study [132], the sample investigated by us was prepared by a novel sol-gel method that allowed precisely controlling the composition and morphology of the product obtained. Importantly, the low sintering temperatures particularly prevent Li loss during the final annealing step.

Rhombohedral Li_1.5Al_0.5Ti_1.5 PO ₄ crystallizes with the space group R\(\bar {3}\)c and adopts the NASICON structure (Na super-ionic conductor, Na_{1 + x} Zr₂Si _x P _3−x O ₁₂ (0<x<3)). It is composed of alternating, corner-sharing [PO₄] tetrahedra and [TiO₆] octahedra in which Li can occupy interstitial sites such as the M1(6b) site (see Fig. 8(c)). If starting from LiTi₂ PO ₄ the Ti⁴⁺ are consecutively replaced by lower valent Al⁺³ ions, for the reason of charge neutrality additional Li ions are inserted that occupy the M3 voids (Fig. 8(c), [134]). This enhancement of the charge carrier concentration directly affects Li ion dynamics. Indeed, at room temperature an already fully motionally narrowed⁷NMR line is recorded that is characterized by a quadrupolar powder pattern with a (mean) coupling constant of 45.4 kHz, see Fig. 8b = (b). The sharp (residual) powder pattern emerges from a featureless quadrupole hump with low intensity seen at very low T.

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Li NMR relaxometry corroborates the high Li diffusivity indicated by line shape measurements; the rates R _{1 diff} and R _{1 ρ diff} when plotted vs the inverse temperature reveal several distinct peaks that point to multiple Li jump processes in Li_1.5Al_0.5Ti_1.5 PO ₄ [125]. The first R _{1 ρ diff}(1/T) peak shows up at ca. 182 K (Fig. 8(a)). Thus, well below ambient temperature the jump rate has already reached values in the order of 10⁵s⁻¹. For comparison, for LLZMO the first maximum, at a similar locking frequency, is seen at 263 K [118]. The corresponding R _{1 diff}(1/T) peak of LATP is found at ca. 333 K. A second pair of relaxation rate peaks is seen at somewhat higher T. The dashed lines in Fig. 8(a) represent so-called joint or global fits that simultaneously reproduce the peak in the laboratory and in the rotating frame. For the two pairs of relaxation processes activation energies were found to be rather small (0.174 eV and 0.160 eV) [125]. Such low values are in agreement with a diffusion pathway that involves interstitial positions as has been the outcome of recent calculations performed by Lang et al. using density functional theory [135].

The differences in T _max might be explained by the pre-exponential factors τ ₀ determining the two underlying Arrhenius relations. Although thermally activated with rather equal activation energies the configurational entropy terms may differ for the two pathways. In LATP Li ions that are (i) displaced from their original sites in Al-free LTP or (ii) trapped in the neighbourhood of Al cations might have access to a different number of available sites to jump to. As in the case of Al-stabilized LLZO, the low-T flank of the R _{1 diff}(1/T) is characterized by an even lower activation energy (0.11 eV) than found with R _{1 ρ diff}(1/T) in the limit ω ₀ τ≫1. While for the rotating-frame data β=2 well reproduces the peak, for those measured in the lab frame the small slope causes β to adopt a value of 1.66. Localized M1-M3 exchange processes or within-site motions in the rather large and distorted M3 voids might be responsible for this behavior [135], which would be in analogy to the caged Li ion dynamics in the split-site 96h-48g-96h in LLZO, see the discussion above.

In Fig. 8(d) the NMR rate peaks of LATP are compared with those of garnet-type c-Li_6.6La₃Zr_1.6Ta_0.4 O ₁₂ [125] and one of the best ion conducting sulfides: Li₆ PS ₅Br, which, in terms of conductivity, is comparable to Li₁₀GeP₂ S ₁₂, see above [35, 62]. Since the locking frequencies used are comparable, the lower the temperature at which the diffusion-induced rate peak shows up, the higher the diffusivity seen via time-domain NMR. Thus, Li ion self-diffusion increases in the order c-Li_6.6La₃Zr_1.6Ta_0.4 O ₁₂ < Li_1.5Al_0.5Ti_1.5 PO ₄ < Li₆ PS ₅Br. While LATP adopts an intermediate position, ultra-fast Li ion exchange is seen for the sulfide [133].

3.5 Sulfides: Li₆ PS ₅Br and Li₇ P ₃ S ₁₁

The NMR rate peaks R _{1 diff}(1/T) and R _{1 ρ diff}(1/T) of Li₆ PS ₅Br are shown in Fig. 9. The rates recorded at a locking frequency of 14 kHz pass through a rate maximum at temperatures as low as 167 K. In the limit ω ₁ τ≫1 the rates turned out to be independent of frequency which points to 3D diffusion. The asymmetric shape of the peaks with its lower slope on the high-T side (ω ₁ τ≪1) clearly reveals correlated motions because of the influence of disorder on the angstrom length scale combined with Coulomb interactions. From a global fit, represented in Fig. 9(b) by the solid line, an activation energy as alow as 0.2 eV is obtained [133]. It represents translational long-range Li ion dynamics; the corresponding value on the low-T side is only 0.08 eV, which is indicative for local(ized) ion dynamics.

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In line with the R _{1 ρ diff}(1/T) data, the rate peaks measured with the saturation recovery sequence show up at temperatures well below ambient. In the limit ω ₁ τ≫1 the rates R _{1 diff} sense the same local hopping as R _{1 ρ diff} does. At T = T _max, with ω ₀/2π=116 MHz we end up with a diffusion coefficient in the order of 7.6×10⁻⁸cm² s⁻¹ at 263 K. This value is obtained if we assume a mean jump distance of ca. 2.5 Å. D translates into a Li ion conductivity in the order of 10⁻²S cm⁻¹ which definably makes Li₆ PS ₅Br an ultra-fast ionic conductor with a liquid-like diffusion behavior [133].

The finding that Li₆ PS ₅Br offers rapid Li ion exchange processes was corroborated by⁷Li NMR line width studies (see Fig. 9(a)) and⁶Li NMR relaxometry measurements in the lab frame, see Fig. 9(b). Due to the weaker electric and magnetic interactions of the⁶Li spins the absolute rates R _{1 diff} are shifted toward lower values. Because of the lower resonance frequency used to acquire the R _{1 diff}(1/T) peak in the case of⁶Li the rate maximum shows up at lower T _max, as expected [133].

As is seen in Fig. 9a = (a), motional line narrowing reveals that while Li self-diffusion in Li₆ PS ₅Br is the fastest, the incorporating of Cl and Br anions in general lead to a jump in Li ion diffusivity. The lower the onset temperature of motional narrowing, the faster Li self-diffusivity. Line narrowing results from averaging of (homonuclear) dipole-dipole interactions due to sufficiently fast hopping processes. At very low temperatures ion diffusivity can be regarded as frozen with respect to the NMR time scale. With increasing temperature, however, Li diffusivity rapidly increases. When mean jump rates in the order of the low-T line width are reached, this is ca. 6 to 7 kHz in the present case (cf. Fig. 9(a)), the line starts to narrow. Therefore, we conclude that at approximately 90 K the jump rate in Li₆ PS ₅Br has already reached values in the order of 10⁴ to 10⁵s⁻¹. For other solid electrolytes, such results are usually seen at 200 K and higher. It is evident that the replacement of S or Se anions with Cl or Br differing in ionic radii causes local strain that may facilitate ionic diffusion [133]. This mismatch in size combined with the different polarizabilities of the halogen ions could serve as an explanation of the enhancement seen when going from Li₇PS(e)₆ to Li₆ PS ₅Br.

As a last example of a fast ion conductor, which plays an increasing role in all-solid state batteries, Li₇ P ₃ S ₁₁ is presented. If at hand in the form of a glass ceramic, the NMR spin-lattice relaxation behavior at both frequencies in the MHz and kHz range turned out to be rather complex. As an example, in Fig. 10 the⁶Li and⁷Li NMR relaxation rates are shown [64]. The rates recorded in the laboratory frame, R ₁, show Arrhenius-type flanks from which a series of activation energies in different ranges of temperature were deduced. For example, the⁷Li NMR rate R ₁ starts to be influenced by dynamic processes at temperatures being larger than 250 K. Then the rate runs through a peak area with unusual shape. Usually the high-T side of a given relaxation peak is expected to be characterized by an activation energy larger than that of the low-T side. In the present case, this is not fulfilled. A similar behavior is seen if the⁶Li nucleus is used to follow spin-lattice relaxation. The anomalous shape of the rate peaks can be explained if we simply assume a number of dynamic phenomena, including also rotational motions of the P-S units, to be responsible for overall longitudinal relaxation.

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Indeed, on the kHz range, even the magnetization transients cannot be represented with a single (stretched) exponential. Only a combination of three decay functions is able to properly describe the decay of M _ρ as a function of locking frequency, see Fig. 10(b). In the upper part of the figure some selected transients M _ρ are shown, in the lower part the transient recorded at 226 K and a locking frequency of 25 kHz is displayed together with the fit used to parameterize its shape; γ _i with i=1,2,3 denote the stretching exponents in \(M_{\rho ,i}(t_{\text {lock}}) \propto \exp (-(t_{\text {lock}}/T_{1\rho })^{\gamma _{i}})\). Interestingly, the fast decay at short locking times t _lock resembles that of spin-spin relaxation behavior. Further NMR experiments making use of³¹P spin-lattice relaxation may throw some light on the dynamic situation in Li₇ P ₃ S ₁₁.

Looking at activation energies from conductivity and impedance spectroscopy, the values deceived from NMR are indeed in agreement with barriers as low as 0.1 eV and 0.2 eV reported for the glass ceramic Li₇ P ₃ S ₁₁, see the overview in ref. [64] and ref. [52]. Interestingly, purely amorphous Li₇ P ₃ S ₁₁ needs, however, to be characterized by larger activation energies in the order of 0.35 eV or higher [136]. For Li ion conductors, usually the glassy (or x-ray amorphous) forms represent better ion conductors than their crystalline counterparts, see, e.g., refs. [137, 138, 139, 140, 141, 142, 143] and ref. [144] for a recent investigation on the thiophosphate Li₄ P ₂ S ₆. Li₇ P ₃ S ₁₁ constitutes an exception. Large amorphous fractions need to be avoided since they seem to hinder the ions to mover easily over long distances. Just recently, a systematic in situ study has been published to find out the optimum synthesis parameters to prepare crystalline Li₇ P ₃ S ₁₁ as a glass ceramic [145].