Isabel
Hanghofer
*^{a},
Bernhard
Gadermaier
^{a},
Alexandra
Wilkening
^{a},
Daniel
Rettenwander
^{a} and
H. Martin R.
Wilkening
*^{ab}
^{a}Institute for Chemistry and Technology of Materials, Christian Doppler Laboratory for Lithium Batteries, Graz University of Technology (NAWI Graz), Stremayrgasse 9, A-8010 Graz, Austria. E-mail: isabel.hanghofer@tugraz.at
^{b}ALISTORE – European Research Institute, CNRS FR3104, Hub de l'Energie, Rue Baudelocque, 80039 Amiens, France. E-mail: wilkening@tugraz.at
First published on 3rd June 2019
High ionic conductivity, electrochemical stability and small interfacial resistances against Li metal anodes are the main requirements to be fulfilled in powerful, next-generation all-solid-state batteries. Understanding ion transport in materials with sufficiently high chemical and electrochemical stability, such as rhombohedral LiZr_{2}(PO_{4})_{3}, is important to further improve their properties with respect to translational Li ion dynamics. Here, we used broadband impedance spectroscopy to analyze the electrical responses of LiZr_{2}(PO_{4})_{3} and Ca-stabilized Li_{1.4}Ca_{0.2}Zr_{1.8}(PO_{4})_{3} that were prepared following a solid-state synthesis route. We investigated the influence of the starting materials, either ZrO_{2} and Zr(CH_{3}COO)_{4}, on the final properties of the products and studied Li ion dynamics in the crystalline grains and across grain boundary (g.b.) regions. The Ca^{2+} content has only little effect on bulk properties (4.2 × 10^{−5} S cm^{−1} at 298 K, 0.41 eV), but, fortunately, the g.b. resistance decreased by 2 orders of magnitude. Whereas, ^{7}Li spin-alignment echo nuclear magnetic resonance (NMR) confirmed long-range ion transport as seen by conductivity spectroscopy, ^{7}Li NMR spin–lattice relaxation revealed much smaller activation energies (0.18 eV) and points to rapid localized Li jump processes. The diffusion-induced rate peak, appearing at T = 282 K, shows Li^{+} exchange processes with rates of ca. 10^{9} s^{−1} corresponding, formally, to ionic conductivities in the order of 10^{−3} S cm^{−1} to 10^{−2} S cm^{−1}.
For their realization, ceramic electrolytes with sufficiently high ionic conductivities are, however, needed.^{5–9} The overall performance of NaSICON-type (Na SuperIonic CONductor) conductors^{10,11} has attracted renewed interest to study the influence of synthesis conditions and dopants on morphology and ion dynamics. Earlier reports have shown that LiM_{2}(PO_{4})_{3} (M: Ge, Ti, Sn, Hf, Zr) exhibits good chemical stabilities under ambient conditions. This property comes along with a wide electrochemical stability window.^{12,13} LiTi_{2}(PO_{4})_{3} is known as a very good Li-ion conductor but it suffers from the fact that Ti(IV) can easily be reduced to Ti(III) when in contact with metallic Li.^{14}
In contrast to LiTi_{2}(PO_{4})_{3}-based ceramics, including those in which Ti^{4+} is partly replaced by Al^{3+}, LiZr_{2}(PO_{4})_{3} (LZP) shows a much better stability against Li anodes. If in contact with a Li metallic anode LZP forms stable decomposing products, such as Li_{3}P and layer-structured Li_{8}ZrO_{6}, that are able to conduct Li ions when present as thin layers. Importantly, this interphase also reveals sufficiently good wetting properties with respect to both Li metal and the electrolyte.^{15} LZP crystallizes with different structures, ionic conductivities sensitively depend on both the overall morphology and the defect chemistry involved. The ionic conductivity can be improved by incorporating different metals such as Y^{3+}, Sc^{3+}, Al^{3+}, La^{3+} or Ca^{2+}; this strategy is usually accompanied by a stabilization of the rhombohedral structure.^{13,16–22} LZP is typically prepared at calcination temperatures of ca. 1173 K, the phase appearing under these conditions is denoted as the α-phase of LZP, which is subdivided in a orthorhombic form (α) and a monoclinic one (α′). At higher sintering temperatures, that is, approximately at 1423 K, the highly conductive rhombohedral structure α-LiZr_{2}(PO_{4})_{3} crystallizing with Rc symmetry is formed. α′-LiZr_{2}(PO_{4})_{3}, obtained at lower temperatures, is usually a mixture of monoclinic LZP and triclinic LZP; its ionic conductivity is reported to be in the order of 10^{−8} S cm^{−1} whereas α-LiZr_{2}(PO_{4})_{3} shows values around 10^{−5} S cm^{−1}.^{15,22–25}
In this work, the structure variation of different educts on the product formed is discussed. We analysed Li ion dynamics in LiZr_{2}(PO_{4})_{3} and Li_{1+2x}Ca_{x}Zr_{2−x}(PO_{4})_{3} (LCZP) by both broadband impedance spectroscopy^{26,27} and ^{7}Li nuclear magnetic resonance (NMR) spectroscopy, i.e., by recording diffusion-induced spin–lattice relaxation rates.^{28–30} The latter are sensitive to both short-range and long-range ion transport through the crystal lattice of LZP. As we deal with powder samples with crystallite diameters in the μm range, NMR rates measured in the laboratory frame of reference are solely sensitive to bulk ion dynamics.
Impedance spectra were recorded with a Novocontrol Concept 80 broadband dielectric spectrometer equipped with a BDS 1200 cell combined with an active ZGS cell (Novocontrol). We measured complex impedances over a frequency range of ten decades (10 mHz to 10 MHz). The temperature in the sample holder was varied from 173 K to 473 K in steps of 20 K; the temperature program was automatically controlled by a QUATRO cryosystem (Novocontrol). During the measurements a dry nitrogen atmosphere was build up around the sample in the cryostat to avoid any contamination with water and/or oxygen.
^{7}Li NMR SLR rates (1/T_{1} = R_{1}) in the laboratory frame were acquired with the well-known saturation recovery pulse sequence. This sequence uses a comb of closely spaced π/2 pulses to destroy any longitudinal magnetization M_{z}. The subsequent recovery of M_{z} was detected as a function of waiting time t_{d} with a π/2 reading pulse: 10 × π/2 − t_{d} − π/2 – acquisition.^{32,33} To construct the magnetization transients M_{z}(t_{d}), we plotted the area under the free induction decays vs. t_{d}. The transients M_{z}(t_{d}) were parameterized with stretched exponentials, M_{z}(t_{d}) ∝ 1 − exp(−(t/T_{1})^{γ}), to extract the rates R_{1}. Additionally, rotating frame ^{7}Li NMR SLRρ rates 1/T_{1ρ} (= R_{1ρ}) were measured by means of the spin lock technique: π/2-p_{lock} – acquisition.^{32} Here, we used a locking frequency ω_{1}/2π of 20 kHz. The duration of the spin-lock pulse t_{lock} was varied from 10 μs to 460 ms. To ensure full longitudinal relaxation between each scan the recycle delay was set to 5 × T_{1}. The R_{1ρ} rates were obtained by analyzing the resulting transients M_{ρ}(t_{lock}) with stretched exponentials with the form M_{ρ}(t_{lock}) ∝ exp(−(t_{lock}/T_{1ρ})^{κ}). The stretching exponent γ varied from 1 to 0.8, the exponent κ ranges from 1 to 0.6.
Finally, mixing time (t_{m}) dependent ^{7}Li SAE NMR decay curves were recorded with the help of the Jeener–Broekaert^{34,35} three-pulse sequence: (90°)X-t_{p}–(45°)Y-t_{m}–45°-acq. We used a constant preparation time t_{p} of 25 μs to acquire two-time sinus–sinus single-spin correlation functions. The mixing time was varied from 30 μs up to several seconds. A suitable phase cycle^{34,36} was employed to suppress unwanted coherences and to eliminate, as best as possible, dipolar contributions affecting the echo that appears after the reading pulse.^{36} Fourier transformation of the spin alignment echoes, starting from the top of the signal, yields ^{7}Li SAE NMR spectra useful to highlight quadrupole intensities due to the interaction of the quadrupole moment of the ^{7}Li spin (spin-quantum number I = 3/2) and a non-vanishing electric field gradient.
Fig. 2 Rietveld refinement of LZP using ZrO_{2} as educt. The sample was sintered at 1150 °C for 20 hours in a closed Al_{2}O_{3}-crucible. LZP crystallizes with space group RcH, see Table S1† for further information. Selected reflections have been indexed with vertical bars. A very small amount (<4 wt%) of ZrO_{2} was detected. The inset shows the ab-plane of the rhombohedral NaSICON structure with the viewing direction along the c-axis. |
The Zr_{2}(PO_{4})_{3} framework of rhombohedral NaSICON-type LZP consists of two ZrO_{6} octahedra and three PO_{4} tetrahedra sharing O atoms. The octahedral and tetrahedral units are alternating with the cations to form infinite chains parallel to the ternary axis of the structure. Each PO_{4} unit shares its oxygen with four ZrO_{6} octahedra of three Zr_{2}(PO_{4})_{3} units to form the NaSICON framework. A 3D network of conduction pathways is formed that is used by the ions to diffuse through the crystal. In this case the Li ions (6b) are octahedrally coordinated by oxygen ions (36f) at the intersection of three conduction channels (A1). The A1 sites are located between pairs of ZrO_{6} octahedra along the c-axis, while the (vacant) interstitial sites (□) A2 can be found between O_{3}ZrO_{3}A1O_{3}Zr-□-O_{3}ZrO_{3}A1. Since the A1–A1 distance in LZP is rather larger, we assume that interstitial sites, such as A2, are involved in Li^{+} diffusion. Because of the large spatial separation of Li ions in LZP, we expect rather low homonuclear dipole–dipole interactions resulting in narrow ^{7}Li NMR lines even in the rigid lattice regime, see below.
In Fig. 2 the result of our Rietveld analysis of the diffraction pattern of LiZr_{2}(PO_{4})_{3} synthesized by using ZrO_{2} as educt is shown. Our structure solution indeed points to rhombohedral symmetry characterized by the space group RcH (ICSD, no. 201935), as already found by the study of Petit et al.^{19}Via high-temperature neutron diffraction (T ≥ 423 K), Catti et al. reported on two Li^{+} positions displaced from the A1 and A2 sites (see Fig. S2†).^{24} Here, in addition to the main rhombohedral phase, a minor impurity of ZrO_{2} (<4 wt%) is seen (see black bars). Rietveld analysis yields the following lattice properties a (= b) = 8.824 Å and c = 22.456 Å; V = 1514.24 Å^{3}. By doping this sample with 5 wt% and 10 wt% Ca^{2+} the cell volume decreases (1512.64 Å^{3} (5 wt% Ca), 1506.22 Å^{3} (10 wt% Ca)) as expected; for further information we refer to Table S1 in the ESI.†
The preparation route with Zr(CH_{3}COO)_{4} as starting material yields almost the same lattice parameters but the amount of non-reacted material is much higher and reaches values as high as 25 wt%. The incorporation of aliovalent Ca^{2+} ions, on the other hand, helps obtaining phase pure LZP with rhombohedral structure, as is illustrated in Fig. 3. The amount of unreacted ZrO_{2} continuously decreases with increasing Ca-content. This behavior is also found for LCZP prepared with the help of Zr(ac)_{4}. Nonetheless, the amount of ZrO_{2} remains much higher (>15 wt%) than that in samples prepared from ZrO_{2} directly. Dots in Fig. 3 denote reflections belonging to ZrO_{2}. Worth noting, here we do not find any additional Bragg reflections that belong to triclinic LZP in all diffractograms. These have been seen in earlier reports.^{22,39–41} We recognize that small variations in synthesis conditions, including heat treatment and cooling rates, might sensitively affect the phase purity of the final compounds.
According to XRPD we decided to study, in detail, ion dynamics of the following samples: LiZr_{2}(PO_{4})_{3} prepared from ZrO_{2} and Zr(ac)_{4} as well as Li_{1+2x}Zr_{2−x}Ca_{x}(PO_{4})_{3} (x = 0.1 and 0.2) prepared with the help of ZrO_{2}. Similar to LiTi_{2}(PO_{4})_{3} and other NaSICON-type materials α-LiZr_{2}(PO_{4})_{3} shows low relative densities, here ranging from 78% to 84%.
In Fig. 4a the inflexion points of the plateaus P1 and P2 are highlighted by filled circles; straight lines connect these circles. The dispersive region belonging to plateau P2 is best seen at 173 K, it can be approximated with Jonscher's power law:^{43}σ′ = σ_{dc} + A_{0}ν^{n}. Here A_{0} is the so-called dispersion parameter and n represents the power law exponent, which takes a value of 0.73 at 173 K. To analyse our data in terms of dimensionality effects, we also studied the frequency dependence of the real part of the complex permittivity; the corresponding isotherms (ε′ vs. ν) are plotted in Fig. 4c; the plateaus P1 and P2 produce a two-step increase of ε′ when coming from high frequencies. As for σ′, also ε′(ν) can be approximated with exponents around 0.75 for P2, see Fig. 4c. Such Jonscher exponents are expected for 3D ionic conduction.^{44}
To determine which capacitances C govern the responses P1 and P2, we used the equation for a parallel-plate capacitor for an estimation C = ε_{0}ε_{r}A/d. Here ε_{0} represents the electric field constant (8.854 × 10^{−12} F m^{−1}), A the area and d the thickness of the sample. While the DC plateau (P2) at high frequencies is characterized by C = 1.2(1) × 10^{−12} F, for the plateau at lower ν we found C = 2.3(2) × 10^{−11} F. Therefore, the plateau associated with C in the pF range represents the bulk response, whereas P1 seen at lower frequencies is additionally governed by grain boundary contributions (g.b.) for which capacitances in the order of 10^{−11} F are typically expected.^{45}
In the Arrhenius plot of Fig. 4b, σ_{DC}T of the two plateaus P1 and P2 is plotted vs. the inverse temperature T. σ_{DC}T does not change when several heating and cooling runs were performed. The lines in Fig. 4b refer to fits according to σ_{DC}T(P1, P2) ∝ exp(−E_{a}/(k_{B}T)); k_{B} denotes the Boltzmann's constant and E_{a} the corresponding activation energy. For LiZr_{2}(PO_{4})_{3} prepared from ZrO_{2} we obtain E_{a}(P2) of 0.41(1) eV (in agreement with ref. 15 (0.40 eV) and ref. 25 (0.39 eV)), whereas for LiZr_{2}(PO_{4})_{3}, when using Zr(ac)_{4} as starting material, E_{a} increases to 0.44(1) eV (see Fig. 4b). At room temperature, the ionic conductivities of P2 are 3.6(2) × 10^{−5} S cm^{−1} and 3.4(2) × 10^{−6} S cm^{−1}, respectively. For the sake of clarity, in Table 1 all activation energies E_{a} are listed, the table also includes values referring to Li_{1+2x}Ca_{x}Zr_{2−x}(PO_{4})_{3} (x = 0.1, x = 0.2) that has been prepared from ZrO_{2}.
composition | E _{a}/eV | log_{10}(σ_{0}/S cm^{−1} K) | Starting material |
---|---|---|---|
a At lower temperatures the activation energy increases to 0.43(4) eV. | |||
LiZr_{2}(PO_{4})_{3} | 0.44(2) (0.48(4)) | 4.97(1) (3.86(2)) | Zr(ac)_{4} |
LiZr_{2}(PO_{4})_{3} | 0.41(2) (0.37(4)^{a}) | 4.91(2) (1.49(7)) | ZrO_{2} |
Li_{1.2}Ca_{0.1}Zr_{1.9}(PO_{4})_{3} | 0.43(1) (0.52(9)) | 5.44(4) (4.14(7)) | ZrO_{2} |
Li_{1.4}Ca_{0.2}Zr_{1.8}(PO_{4})_{3} | 0.41(2) (0.38(4)) | 5.08(2) (3.49(1)) | ZrO_{2} |
Unfortunately, by using Nyquist diagrams (see Fig. S3†), which show complex plane plots of the imaginary part Z′′ of the complex impedance, Z, vs. its real part Z′, the faster relaxation process, corresponding to P2, cannot be resolved properly. Instead, we used the complex modulus M′′(ν) representation^{46} to visualize the two processes, see Fig. 5a and 6 (Fig. 6b includes data for Li_{1+2x}Ca_{x}Zr_{2−x}(PO_{4})_{3} prepared from ZrO_{2}).
Fig. 6 (a) Frequency dependence of the imaginary part M′′ of the complex modulus, the real part, ε′, of the permittivity as well as the real part, σ′, of the complex conductivity of LiZr_{2}(PO_{4})_{3} (prepared from ZrO_{2}). The isotherm was recorded at 213 K. The maxima in M′′ clearly refer to the plateaus P1 and P2 seen in σ′(ν). For comparison, the change of ε′(ν) is shown, too. (b) Arrhenius plot of the ionic conductivities referring to the plateaus P1 and P2 (σ_{DC}T vs. 1000/T) of cation-mixed crystalline Li_{1+2x}Zr_{2−x}Ca_{x}(PO_{4})_{3}. Solid lines and dashed lines represent fits according to an Arrhenius law yielding the activation energies E_{a} as indicated. While solid lines refer to conductivities influenced by the grain boundary contributions, the dashed lines represent bulk ion dynamics in Li_{1+2x}Zr_{2−x}Ca_{x}(PO_{4})_{3}. For the sake of clarity, the bulk ion conductivities for the samples with x = 0.20 and x = 0.10 have been plotted using an offset of +2 and +1.5 on the logarithmic scale. These conductivities coincide with those of the Ca-free sample. Activation energies with error margins are listed in Table 1. |
At sufficiently low temperatures two relaxation peaks M1 and M2 appear (see the orange arrows in Fig. 5a). These peaks are separated by two orders of magnitude on the frequency scale. This distance on the frequency scale is comparable with the ratio of σ_{DC,P2}:σ_{DC,P1}; thus, they refer, as is seen in Fig. 6 to the plateaus P1 and P2 governing σ′(ν). As expected the two electrical relaxation frequencies mirror the ratio in conductivities. The peak with the larger amplitude (M2) corresponds to P2 in σ′(ν), the one with the smaller amplitude (M1) represents a relaxation process with a longer relaxation time (cf. P1), see also Fig. 6. As an estimation, M′′ is proportional to the inverse capacitance, M′′ ∝ 1/C.^{45,47} Thus, we expect peak M2 to be characterized by a larger amplitude (3.4 pF) than M1 (30.1 pF). C_{1}/C_{2} ≈ 10 is in good agreement with the amplitude ratio seen in Fig. 5a and 6a.
In order to compare activation energies extracted from σ_{DC}(P1, P2) we determined characteristic electrical relaxation frequencies 1/τ_{M} using the modulus peaks of Fig. 5a. 1/τ_{M} refers to frequencies at which the peaks appear. A comparison of activation energies for selected compounds is shown in Fig. 5b; errors are, at least, in the order of ±0.01 eV.
As mentioned in the beginning, Ca^{2+} incorporation increases the ionic conductivity of LZP. Fig. 6b shows the change in σ_{DC}T(P1, P2) of Li_{1+2x}Ca_{x}Zr_{2−x}(PO_{4})_{3} prepared from ZrO_{2} for x = 0, x = 0.1 and x = 0.2. Most importantly, while σ_{DC}T(P2), which refers to bulk ion dynamics, is only slightly affected by x (E_{a} ranges from 0.41 eV to 0.43 eV, see Fig. 6b and Table 1 that also includes the prefactors), Ca^{2+} incorporation mainly reduces the g.b. resistance. We clearly see that σ_{DC}T(P1) is by two orders of magnitude larger than that of the sample with x = 0, see Fig. 6b. Most likely, the sintering process benefits from a Ca-rich composition, which helps reducing the blocking nature of surface regions of the Li_{1+2x}Ca_{x}Zr_{2−x}(PO_{4})_{3} crystallites. For x = 0.2 we obtain σ_{DC}(P2) = 4.2 × 10^{−5} S cm^{−1} at 293 K. This value is only slightly lower than that of Li et al.^{15} (3.8 × 10^{−5} S cm^{−1}) when samples are compared that have been prepared by conventional sintering. If pellets were fired by spark plasma sintering, bulk values of 1.8 × 10^{−4} S cm^{−1} were reported.^{15} It would be interesting to see, in coming studies, whether Ca^{2+}, here acting as a sintering aid, segregates in the g.b. regions; such segregation would explain the finding that ion transport in the bulk regions is only little affected by Ca^{2+} incorporation.
Careful inspection of σ_{DC}T(P1) reveals that, if the samples with x = 0.2 is considered as an example (Fig. 6b), a slight kink is seen for temperatures well above room temperature; the activation energy changes from 0.48 eV to 0.38 eV at higher T. Presumably, Ca^{2+} in the grain boundaries influences the Li^{+} arrangements in these regions. Li^{+} order/disorder phenomena might lead to such changes.
Fig. 7 (a) Arrhenius plot of the ^{7}Li NMR relaxation rates R_{1} and R_{1ρ} of Li_{1.4}Ca_{0.2}Zr_{1.8}(PO_{4})_{3} (prepared via the route using ZrO_{2}) measured in the laboratory frame of reference (116 MHz) and in the rotating frame of reference (20 kHz, nominal locking frequency). Dashed lines represent BPP-type fits to determine activation energies, prefactors and asymmetry parameters β. Temperatures indicate T_{max} of the rate peaks. The superposition of the R_{1} rate peaks can be approximated with two symmetric peaks (β = 2). (b) Li^{+} jump rates as deduced from the diffusion-induced rate peaks R_{1} and R_{1ρ}. For comparison, we also included jump rates that we estimated from frequency-dependent conductivity measurements and ^{7}Li SAE NMR. See text for further explanations and Table 2 for error margins of the activation energies shown. |
Below 183 K the rates R_{1} reveal a non-diffusive background regime. In this temperature range longitudinal relaxation is induced by lattice vibrations or coupling of the Li spins with paramagnetic impurities.^{48–50} At higher temperatures we expect the SLR rate to be increasingly induced by Li^{+} hopping processes. Such processes lead to magnetic and electric field fluctuations that cause longitudinal relaxation.^{48} Indeed, the rates increase with temperature and, in both cases R_{1} and R_{1ρ}, characteristic diffusion-induced rate peaks appear. Importantly, we recognize that R_{1} passes through two maxima located at T_{max} = 282 K and 492 K, respectively. In general, at T_{max} the motional correlation rate 1/τ_{c} is related to ω_{0}via the relation τ_{c}ω_{0} ≈ 1. The so-called motional correlation rate 1/τ_{c} is identical, within a factor of two, with the Li^{+} jump rate 1/τ.^{48,50} For R_{1ρ}, this maximum condition changes to τ_{c}ω_{1} ≈ 0.5.^{51} As ω_{0} and ω_{1} differ by more than three orders of magnitude, we are able to characterize Li^{+} motional correlation rates in LCZP with values in both the kHz and MHz range.^{52}
Here, we approximated the superposition of the two R_{1} rate peaks by a sum of two Lorentzian-shaped spectral density functions J(ω_{0}, T) ∝ τ_{c}/(1 + (ω_{0}τ_{c})^{β}) according to the concept introduced by Bloembergen, Purcell and Pound (BPP) for 3D isotropic diffusion,^{53,54} see dashed line in Fig. 7a that follows the R_{1} rates. For each peak we used a single term to deconvolute the temperature dependence of the overall R_{1} rates measured, cf. the dashed-dotted lines in Fig. 7a, which will be discussed below. The rate 1/τ_{c} usually obeys the Arrhenius relation 1/τ_{c} = 1/τ_{c,0}exp(−E_{a}/(k_{B}T)); 1/τ_{c,0} represents the pre-exponential factor that is typically identified as the “attempt frequency” of the jump process.^{55}
In general, J(ω_{0}, T) is the Fourier transform of the underlying motional correlation function G(t′).^{48} If G(t′) is or can be well approximated with a single exponential, β equals 2. Values smaller than 2 are expected for correlated motion, which is, e.g., seen for cations exposed to an irregularly shaped potential landscape. In such a landscape short-ranged Li^{+} diffusion will be different to long-range ion transport. In particular, forth-and-back jumps or, more generally speaking, localized motions will govern the rate R_{1} in the low-temperature regime, which is characterized by τ_{c}ω_{0} ≫ 1. In this regime we have J(ω_{0}) ∝ τ_{c}^{−1}ω_{0}^{−β} with (1 < β ≤ 2). β < 2 produces asymmetric rate peaks which are often found for structurally complex ion conductors with a non-uniformly shaped energy landscape. Ion dynamics in this regime are anticipated to be affected by correlation effects because of both structural disorder and strong Coulomb interactions of the moving ions.^{56} As the peak is asymmetric, the activation energy on this side of the peak, E_{a, low}, is lower than that of the high-temperature flank, E_{a, high}; the two values are linked to each other via E_{a, low} = (β_{(ρ)} − 1)E_{a, high}. In the regime τ_{c}ω_{0} ≪ 1, that is, on the high-temperature side of the peak, many jump events are sensed during one Larmor precession and the probability is high that also these jumps contribute to longitudinal relaxation which are characterized by higher activation energies. Usually, on this side of the rate peak R_{1}(1/T) long-range Li ion dynamics is sensed; in this limit we obtain J(ω_{0}) ∝ τ_{c}.^{29}
In the present case, approximating the rate peaks with a sum of two BPP-type spectral densities yields activation energies of E_{a,1} = 0.210(5) eV and E_{a,2} = 0.143(5) eV. As both rate peaks join up, information on β are difficult to obtain. Here, the fitting routine yields β = 2 for the two peaks resulting in symmetric peaks with E_{a, low} = E_{a, high} = E_{a,i} (i = 1, 2). Note that the peak at higher T (i = 2) is only partly visible which influences the precise determination of E_{a, high}. We clearly recognize that E_{a,i} as determined from R_{1} SLR NMR turned out to be significantly lower than E_{a} obtained from σ_{DC} measurements describing bulk ion dynamics. The fact that E_{a,i} < E_{a, DC} shows that the number of jump events seen by NMR does not include all types of jumps needed for long-range diffusion. Obviously, the spin-fluctuations sensed by NMR are already sufficient to generate a full R_{1} peak. Thus, we conclude that the Li^{+} ions in LCZP are highly mobile on a short-range length scale while long-range ion transport is, however, characterized by much larger activation energies than 0.2 eV. This situation resembles that of Li^{+} ion dynamics in argyrodite-type Li_{6}PS_{5}I, which has been studied recently by our group.^{42} Here, we assume that rapid forward–backward exchange processes between the sites A1 and A2 might be responsible for the peaks seen in NMR spin–lattice relaxometry. A2 sites might be occupied in samples with x > 0. Here, especially the peak appearing at T_{max} = 282 K for Li_{1.4}Ca_{0.2}Zr_{1.8}(PO_{4})_{3} points to rapid (localized) exchange processes with residence times in the order of several ns. According to ω_{0}τ_{c} ≈ 1 we estimate that at T_{max} the jump rate should be in the order of 1/τ = 7.3 × 10^{8} s^{−1} ≈ 10^{9} s^{−1}. A very similar behavior has recently been seen also for NaSICON-type Na_{3.4}Sc_{0.4}Zr_{1.6}(SiO_{4})_{2}PO_{4}.
To complement our R_{1} measurements, we carried out spin-lock NMR SLR measurements at a locking frequency of 20 kHz. As expected we detected a prominent spin-lock NMR peak R_{1ρ}(1/T) at much lower temperature than 492 K. A single peak appears at T_{max} = 293 K. It turned out to be slightly asymmetric with an activation energy of E_{a, high} = 0.185(5) eV and E_{a, low} = 0.168(5) eV. At first glance we would say that the R_{1ρ} peak might correspond to the R_{1} peak seen at T_{max} = 492 K. Keeping, however, both the locking frequency of only 20 kHz and the rather low activation energy of 0.185 eV in mind, the R_{1ρ} peak belonging to R_{1}(1/T) with T_{max} = 492 K would be expected to appear at much lower temperatures than ambient. Even if we replace ω_{1} by an effective frequency ω_{1, eff} (>ω_{1}), which takes local magnetic fields into account that increase ω_{1}, no satisfactory joint fit results that is characterized by the same E_{a} and the same τ_{c,0} for the two peaks. Here, only unreliably high ω_{1, eff} values reaching the MHz range would result in a joint fit connecting the two peaks. Table 2 shows an overview of the results obtained from analyzing the three peaks individually by BPP-type spectral density functions. It also includes the amplitudes C_{NMR} in R_{1(ρ)} = C_{NMR}J(ω_{0}, T). We see that the amplitudes of the two R_{1} rate peaks differ by approximately one order of magnitude. Most likely, stronger quadrupolar relaxation governs the peak appearing at higher T. The corresponding prefactor 1/τ_{0} is relatively low, while that of the peak showing up at 282 K (1/τ_{0} = 5.9 × 10^{12} s^{−1}) is consistent with frequencies typically expected for phonons.
E _{a} (= E_{a, high}) | C _{NMR} | β _{(ρ)} | τ _{0} | |
---|---|---|---|---|
a The value in brackets refers to E_{a, low} of the R_{1ρ} peak seen at 293 K. | ||||
R _{1} (T_{max} = 492 K) | 0.143(5) eV | 1.8(1) × 10^{11} s^{−2} | 2 | 4.1(2) × 10^{−11} s |
R _{1} (T_{max} = 282 K) | 0.210(5) eV | 2.5(1) × 10^{10} s^{−2} | 1.93 | 1.7(2) × 10^{−13} s |
R _{1,ρ} (T_{max} = 293 K) | 0.185(5) eV | 1.5(1) × 10^{9} s^{−2} | 2 | 2.8(1) × 10^{−9} s |
(0.168(5) eV)^{a} |
The above-mentioned jump rate derived from SLR NMR at T_{max} (1/τ = 7.3 × 10^{8} s^{−1}) can be converted into diffusion coefficients with the help of the Einstein–Smoluchowski equation according to D_{NMR} = a^{2}/(6τ), which is valid for 3D diffusion.^{29} As an estimation, by inserting a = 6.3 Å, which is simply the shortest Li–Li distance, we obtain D_{NMR} = a^{2}/(6τ) = 4.8 × 10^{−11} m^{2} s^{−1}. Assuming the interstitial sites A2 participating in exchange processes D_{NMR} reduces to 1.8 × 10^{−11} m^{2} s^{−1}. A distance of a = 6.3 Å is, of course, by far too long for a single hopping process. In the structural model of Catti et al. the distance reduced to 4.2 Å at 423 K.^{24}
In the Arrhenius plot of Fig. 7b we compare the rates 1/τ from NMR (1/τ_{NMR}) with those obtained after converting σ_{DC} into jump rates by using the Nernst Einstein equation. As we do not know the exact jump distance, we calculated 1/τ_{σ} for three different values of a. 1/τ_{σ} represents an average value mainly influenced by Li^{+} jumps necessary for long-range ion transport. Most likely, the R_{1} peak appearing at 492 K is more related to jump processes enabling the ions to move over longer distances. The large discrepancy between 1/τ_{NMR}(282 K) and 1/τ_{σ} once again visualizes that the R_{1}(1/T) peak at low T reflects localized jump processes. We do not find evidences that this peak originates from any phase transitions, as the phase-pure Ca-bearing samples show rhombohedral structure. For comparison, in Fig. 7b we also included 1/τ_{σ} rates estimated from conductivity values that characterize the influence of grain boundaries. Interestingly, 1/τ_{NMR}(293 K, R_{1ρ}) = 2.5 × 10^{5} s^{−1}, which we estimated via the relation ω_{1}τ_{c} ≈ 0.5, see above, agrees with 1/τ_{σ, g.b}. In contrast to R_{1}, which as far as μm-sized crystallites are considered, is mainly affected by bulk processes, the corresponding spin-lock NMR rates R_{1ρ} seems to be (also partly) sensitive to ion dynamics also influenced by interfacial processes. This behaviour is consistent with the following comparison. At frequencies in the kHz range (ω_{1}/2π = 20 kHz) and at T = 282 K the isotherms σ′(ν) are mainly influenced by the g.b. response rather than bulk ion dynamics.
To shed more light on long-range ion dynamics, we used ^{7}Li NMR line shape measurements and ^{7}Li SAE NMR experiments^{46,57–60} to further characterize ionic transport in LCZP. Via SAE NMR we should be able to get access to diffusion parameters that characterize ion transport over longer distances as the method is sensitive to exchange processes on the time scale that is comparable to that of DC conductivity measurements.^{48} In Fig. 8a variable-temperature ^{7}Li NMR spectra of Li_{1.4}Ca_{0.2}Zr_{1.8}(PO_{4})_{3} are shown. Remarkably, at temperatures as low as 213 K a relatively narrow NMR line is detected whose width is only 1.4 kHz. Usually, we would expect a width in the order of several kHz due to ^{7}Li–^{7}Li dipolar interactions. Here, the large Li–Li distance of 6.3 Å between the A1 sites, and between A1 sites and Li ions occupying interstitial sites in samples with x > 0, leads to relatively weak dipole–dipole interactions producing a narrow line already in the rigid-lattice regime. With increasing temperature, the line undergoes a slight narrowing process because of Li diffusion that averages dipolar couplings. Finally, at even higher temperatures, i.e., in the extreme narrowing regime, its width is only governed by the inhomogeneity of the external magnetic field.
Interestingly, at 313 K two satellite lines emerge that belong to a first-order quadrupole powder pattern arising from the interaction of the quadrupole moment of the ^{7}Li nucleus (I = 3/2) with a non-vanishing electric field gradient (EFG) at the nuclear site. The EFG is produced by the electric charge distribution in the direct neighborhood of the ^{7}Li nucleus. This additional interaction alters the Zeeman levels and, thus, also the associated (angular) Zeeman frequency ω_{0} towards ω_{0} ± ω_{Q}.^{32,54} The singularities seen in Fig. 8a correspond to the 90° satellite transitions of a powder pattern, which typically show up at sufficiently high T; this feature belongs to the universal characteristics of crystalline materials studied by NMR.^{32,50,61} Their distance on the frequency scale, if we simply assume an EFG with axial symmetry, corresponds to δ/2. δ is the quadrupole coupling constant. A distance of 49 kHz leads to δ = 98 kHz,^{58} which is in fair agreement with the value reported by Petit et al.^{20} Here, this value should, however, be interpreted as an average value, as we cannot exclude a small distribution of EFGs the ions are exposed to at very low temperatures. At very low T, the intensity of the singularities is too low to be detectable by single pulse experiments. Instead, echo experiments should be used that are able to avoid receiver dead time effects. In Fig. 8b (see inset) a spectrum is shown that is the Fourier transform of a stimulated echo. Indeed, a sharp central line is located on top of a broad quadrupole foot. Hence, we conclude that the Li ions are exposed to a distribution of EFGs.
Fluctuations in ω_{Q} seen by the ions when jumping between electrically inequivalent sites can be used to record sinus–sinus two-time correlation functions. In Fig. 8a the change of the ^{7}Li SAE amplitude S_{2} is shown vs. the logarithmic mixing time t_{m}. S_{2} depends on both the preparation time t_{p} and t_{m}. Here, we measured the decay curve at fixed t_{p} (= 20 μs) but variable mixing time. Stretched exponentials of the form S_{2} ∝ exp(−(t_{m}/τ_{SAE})^{λ}), with a stretching factor λ ranging from 0.21 to 0.37, are best suited to describe the dependence of S_{2}(t_{p} = const., t_{m}) in this temperature regime. In general, stretching factors deviating from λ = 1 indicate non-Debye-like motional process. For example, such deviations can arise from motions in disordered matrices or in confined dimensions^{59,62} leading to motional correlation functions whose decay slows down with increasing observation time. With increasing T the inflexion point of the echo decay curves shifts towards shorter t_{m}. At the same time, the shape of S_{2} steadily becomes more stretched until a value of λ = 0.21 is reached at T = 203 K. At sufficiently long mixing times the curves S_{2}(t_{p} = const., t_{m}) always reach S_{2,∞} = 0, which either indicates a rather large number of quadrupole frequencies involved or which points to the influence of dipolarly coupled spins, as is well-described in literature.^{36} The rates 1/τ_{SAE} governing the stretched decay functions are included in Fig. 7b. We recognize that they are in fair agreement with those rates, 1/τ_{σ, bulk}, which were estimated from bulk ionic conductivities of Li_{1.4}Ca_{0.2}Zr_{1.8}(PO_{4})_{3}.
Footnote |
† Electronic supplementary information (ESI) available: Further X-ray powder patterns, results from structure solution and Nyquist plots. See DOI: 10.1039/c9dt01786k |
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