I.
Hanghofer
*^{a},
M.
Brinek
^{a},
S. L.
Eisbacher
^{a},
B.
Bitschnau
^{b},
M.
Volck
^{c},
V.
Hennige
^{c},
I.
Hanzu
^{ad},
D.
Rettenwander
^{a} and
H. M. R.
Wilkening
*^{ad}
^{a}Christian Doppler Laboratory for Lithium Batteries and Institute for Chemistry and Technology of Materials, Graz University of Technology (NAWI Graz), Stremayrgasse 9, 8010 Graz, Austria. E-mail: isabel.hanghofer@tugraz.at; wilkening@tugraz.at; Fax: +43 316 873 32332; Tel: +43 316 873 32330
^{b}Institute of Physical and Theoretical Chemistry, Graz University of Technology, Stremayrgasse 9, 8010 Graz, Austria
^{c}AVL List GmbH, 8020 Graz, Austria
^{d}Alistore-ERI European Research Institute, 33 rue Saint Leu, 80039 Amiens, France
First published on 27th March 2019
For the development of safe and long-lasting lithium-ion batteries we need electrolytes with excellent ionic transport properties. Argyrodite-type Li_{6}PS_{5}X (X: Cl, Br, I) belongs to a family of such a class of materials offering ionic conductivities, at least if Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl are considered, in the mS cm^{−1} range at room temperature. Although already tested as ceramic electrolytes in battery cells, a comprehensive picture about the ion dynamics is still missing. While Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl show an exceptionally high Li ion conductivity, that of Li_{6}PS_{5}I with its polarizable I anions is by some orders of magnitude lower. This astonishing effect has not been satisfactorily understood so far. Studying the ion dynamics over a broad time and length scale is expected to help shed light on this aspect. Here, we used broadband impedance spectroscopy and ^{7}Li NMR relaxation measurements and show that very fast local Li ion exchange processes are taking place in all three compounds. Most importantly, the diffusion-induced NMR spin–lattice relaxation in Li_{6}PS_{5}I is almost identical to that of its relatives. Considering the substitutional disorder effects in Li_{6}PS_{5}X (X = Br, Cl), we conclude that in structurally ordered Li_{6}PS_{5}I the important inter-cage jump processes are switched off, hindering the ions from taking part in long-range ion transport.
To take full advantage of all-solid-state Li (or Na) batteries, electrolytes with exceptionally high ionic conductivities are needed.^{8–15} In the past, only a few ceramics were available fulfilling this requirement.^{16,17} This lack of suitable materials prevented any commercialization on a large scale.^{18,19} Ideal candidates should (i) show room-temperature ionic conductivities with values comparable to those of liquids, (ii) be chemically stable over a sufficiently wide temperature range and (iii) possess a sufficiently high electrochemical stability over a large potential window.^{10,11} For battery applications, the conductivity of a ceramic material should reach a value of 10^{−3} S cm^{−1} at room temperature.^{10,14}
Over the last years several classes of materials have been presented whose members show very high conductivities. These materials include sulfides, such as Li_{7}P_{3}S_{11} and Li_{10}GeP_{2}S_{12}, oxides including garnets, such as Li_{7}La_{3}Zr_{2}O_{12}, or phosphates based on the NASICON structure.^{8–15,20–26} Although the ionic bulk conductivities of oxides and phosphates can reach very high values, the materials may suffer from high grain boundary (g.b.) resistances.^{27} For the softer sulfides the difference between the bulk and g.b. conductivities^{28} is expected to be much lower. Of course, Li-bearing sulfides or thiophosphates are sensitive to air and moisture, thus requiring careful handling and adequate processing. As an example, the Ge-containing thiophosphate Li_{10}GeP_{2}S_{12} (LGPS),^{13} which initiated the renaissance to consider sulfide-based systems as candidates for solid electrolytes, is characterised by an ionic conductivity of 12 mS cm^{−1}. Meanwhile the expensive element Ge has been replaced by cheaper and naturally abundant materials.^{29} The electrochemical stability of LGPS-based systems needs, however, to be improved.^{30}
Besides LGPS, Deiseroth and co-workers introduced Li-containing argyrodite-type Li_{6}PS_{5}X (X = Cl, Br, I).^{31} The crystal structure is shown in Fig. 1; the possible Li exchange processes are illustrated as well. As an example, the ionic conductivity of Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl is reported to range from 10^{−3} S cm^{−1} to 10^{−2} S cm^{−1} depending on the preparation technique, which influences the type and number of defects as well as the overall morphology.^{32,33} Thus, the ionic conductivity of Li-argyrodites is only slightly smaller than that of the LGPS family. Lower costs make this family of electrolytes highly attractive for application in all-solid-state batteries.^{31,34}
In 2008 Deiseroth et al. reported the solid-state synthesis of Li_{6}PS_{5}X (X = Cl, Br, I) and characterised the local and long-range structure by X-ray diffraction and NMR.^{31} Three years later Rao et al. discussed the synthesis of the lithium argyrodite via ball milling.^{35} Rayavarapu et al. showed that the disorder in both the lithium distribution over the available cation sites and the disorder in the S^{2−}/Cl^{−} and S^{2−}/Br^{−} sublattices promotes lithium ion conductivity.^{36} This view has further been supported by bond-valence calculations.^{37}
Over the last years a number of studies have been carried out to measure the ionic conductivities of the Li_{6}PS_{5}X family by using standard impedance measurements.^{31–40} The overall goal, not only for Li_{6}PS_{5}X, is to relate macroscopic application relevant properties to crystal chemical and morphological properties. So far, the results reported have rarely been related to the length-scale specific properties of the measuring technique applied. The ionic conductivities for Li_{6}PS_{5}X vary from 10^{−7} S cm^{−1} to 10^{−2} S cm^{−1}, while the highest values are found for X = Cl, Br. The corresponding activation energies reported, either experimentally probed or calculated, take values from 0.11 eV to 0.57 eV.^{31–40} Astonishingly, the I-compound Li_{6}PS_{5}I shows the lowest conductivity. According to the strategy of substitutional disorder, size mismatch and polarizability of the halogen anion one could have expected the opposite trend. Replacing S by a larger and more polarizable anion such as I, as compared to the small and unpolarizable Li cation, is expected to cause lattice distortions which on their part lead to a broader distribution of slightly inequivalent Li sites in the Li sublattice. Without any strong site preference of the Li ions they are expected to jump quickly between the local minima of such a heterogeneous, i.e., non-uniformly shaped, potential landscape. Such a situation, sometimes also connected to geometric frustration, is very similar to F diffusion in the recently investigated system (Ba,Ca)F_{2},^{41,42} Li diffusion in Li_{7}La_{3}Zr_{2}O_{12} oxides^{43} and Na diffusion in some closo-borates.^{44} While this situation might be the origin for the high conductivity in Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl, for the I analogue it seems to incompletely describe the real situation as for Li_{6}PS_{5}I relatively low ion conductivities are reported.
In an elegant way, Kraft et al.^{34} tried to establish a connection between the stiffness (or polarizability) of the anion sublattice in Li_{6}PS_{5}X and the macroscopic ionic transport properties. They used impedance spectroscopy and ultrasonic speed of sound measurements to collect information on activation energies and the pre-factors governing the underlying Arrhenius equation. They reported that soft lattices, as is the case for Li_{6}PS_{5}I, lead to smaller Arrhenius pre-factors than obtained for Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br. This decrease in pre-factor, which might originate from a change of the migration entropy or attempt frequencies, has been used to explain the low (overall) ion conductivity in Li_{6}PS_{5}I.
Here, we employed impedance and NMR spectroscopy to shed more light on the effect seen for Li_{6}PS_{5}I. The important structural differences between the three argyrodites have to be taken into account to explain the significant changes in ionic conductivities, pre-factors and activation energies. For this purpose we have not only looked at the monosubstituted compounds like Li_{6}PS_{5}X, but also at samples with two or three different anions such as Li_{6}PS_{5}Cl_{0.5}Br_{0.5} or Li_{6}PS_{5}Cl_{1/3}Br_{1/3}I_{1/3}. The non-substituted compound Li_{7}PS_{6} served as a reference material. Importantly, complementary techniques which are sensitive to ion dynamics on different length-scales and time-scales are used to fully characterise the transport properties. We will show that, on a short-range length scale, the Li ions in Li_{6}PS_{5}I are as rapid as in the Br and Cl analogues. Obviously, the ordered S^{2−}/I^{−} sublattice, as compared to the disordered situation for Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br, is responsible for a more regular potential landscape with a higher degree of site preference for the Li ions. As a consequence, we assume that intercage jump processes in Li_{6}PS_{5}I occur less frequently. These processes are, however, needed to establish rapid long-range ion dynamics in the iodide.
Fig. 2 (a) X-ray powder diffraction patterns of the Li_{6}PS_{5}X series investigated within this study (X = Cl, Cl_{0.5}Br_{0.5}, Br, Br_{0.5}I_{0.5}, I and Cl_{0.33}Br_{0.33}I_{0.33}). The XRPD patterns were recorded after the final annealing step at room temperature. In some cases, little, almost negligible, amounts of Li_{3}PO_{4} and LiX are visible; the amount of LiX does not exceed 1.5 wt%. Only for Li_{6}PS_{5}Cl ca. 3.3 wt% Li_{3}PO_{4} is seen, see the arrows; in all other cases its amount is negligible or less than 1.5 wt%, see the ESI† for Rietveld refinements (Fig. S1 and S2). (b) Lattice parameter of Li_{6}PS_{5}X obtained from Rietveld refinements of the patterns shown in (a). The lattice parameter increases continuously with the radius of X. |
Fig. 3 Rietveld analysis of the XRPD pattern of Li_{6}PS_{5}Br recorded at 293 K (R_{Bragg} = 4.633%). In this plot the Bragg positions as well as calculated and observed profiles are depicted. The analysis reveals a phase pure sample. The R_{w} profile is 10.84% and the GoF amounts to 4.82; see the ESI† for further explanations. |
^{7}Li NMR SLR rates 1/T_{1} ≡ R_{1} in the laboratory frame were acquired with the well-known saturation recovery pulse sequence.^{45,46} 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.^{15} To construct the magnetization transients M_{z}(t_{d}), we plotted the area under the free induction decays (FIDs) vs. t_{d}. Up to 16 FIDs were accumulated for each waiting time.
The transients M_{z}(t_{d}) were parameterised with stretched exponentials, M_{z}(t_{d}) ∝ 1 − exp(−(t_{d}/T_{1})^{γ}), to extract the SLR rates R_{1}. Additionally, rotating-frame ^{7}Li NMR SLRρ rates 1/T_{1ρ} ≡ R_{1} were measured by means of the spin-lock technique: π/2 − p_{lock} acquisition. We used a locking frequency ω_{1}/2π of 20 kHz.^{47,48} The duration t_{lock} of the locking pulse p_{lock} was varied between 22 μs and 460 ms. To ensure full longitudinal relaxation between each spin-lock scan the recycle delay was set to at least 5 × T_{1}. Again, the R_{1ρ} rates were obtained by analysing the resulting transients M_{ρ}(t_{lock}) with stretched exponentials of the form M_{ρ}(t_{lock}) ∝ exp(−(t_{lock}/T_{1ρ})^{κ}). While the stretching exponents γ varied from 1 to 0.9, the exponents κ range from 0.4 to 1, depending on the temperature. Some of the T_{1ρ} transients showed bi-exponential behaviour and were analyzed with a sum of two stretched exponential functions.
In Fig. 2a the XRPD patterns of all samples are shown including a reference pattern taken from the literature (entry no. 418490 in the inorganic crystal structure database (ICSD)). The positions and intensities of the reflections obtained for the different patterns of Li_{6}PS_{5}X including Li_{7}PS_{6} match very well with what is expected from the literature. Humps at low diffraction angles originate from the mercapto foil used to protect the samples from reaction with air during the measurements.
To characterise the samples in detail, Rietveld refinements were carried out. As an example, the result of the analysis is shown for Li_{6}PS_{5}Br in Fig. 3. The Li_{6}PS_{5}Br sample is phase pure and crystallises with cubic symmetry (space group F4m); the lattice parameter turned out to be a = 9.986 Å. The same symmetry is found for the other samples; the amount of side phases or impurities turned out to be extremely low, see the ESI.† When going from Li_{6}PS_{5}Cl to Li_{6}PS_{5}I the lattice parameter a increases from a = 9.857 Å to a = 10.145 Å. Lattice expansion of the unit cell is expected because the radius r of the anion increases in the order r_{Cl} < r_{Br} < r_{I}. Here, an increasing unit cell volume stabilises the lithium ions in the transition state 24g; the change in lattice constant a seems to also affect the Li occupancies in the Li_{6}PS_{5}X series as has been shown by neutron diffraction.^{34}
Recent neutron and X-ray synchrotron diffraction studies have shown that in fc cubic argyrodite-type Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br the S^{2−} and X anions can occupy three different crystallographic positions. Whereas the 16e site is fully occupied by S^{2−} anions, the sites 4a and 4d are shared by the S^{2−} and X anions. For Li_{6}PS_{5}Cl Kraft et al. found that 38.5% of the 4a site and 61.5% of the 4d site is occupied by Cl^{−}.^{34} The rest is filled by S^{2−} anions; Li^{+} only occupies the 48h sites. For Li_{6}PS_{5}Br Rietveld analysis yielded occupancies of 77.9% (4a) and 22.1% (4d) with Li^{+} distributed over the 48h (44.1%) and 24g (11.9%) sites. Thus, the highest degree of anion disorder is found for Li_{6}PS_{5}Cl, while cation disorder for the samples rich in Br. This finding is in contrast to what Rietveld refinement resulted in for Li_{6}PS_{5}I. While Li disorder is also present for the split-site (48h (39.1%) and 24g (21.9%)), in Li_{6}PS_{5}I the I anions solely occupy the 4a sites. The 4d sites are fully occupied by S^{2−}. Deiseroth and co-workers also reported on the same order/disorder effects earlier.^{31,40,49}
The larger difference in ionic radii between I^{−} and S^{2−} (r_{I−} = 216 pm vs. r_{S2−} = 184 pm, as compared to r_{Cl−} = 181 pm and r_{Br−} = 195 pm) might lead to this site preference for the anions. The anion sublattice in Li_{6}PS_{5}I is, thus, structurally ordered as compared to Li_{6}PS_{5}Cl (and Li_{6}PS_{5}Br); simultaneously, cation disorder shows up for Li_{6}PS_{5}Br and Li_{6}PS_{5}I.^{36,37,50} The compounds with mixed halogen compositions agree well with the trend in S^{2−}/X occupancy and cation disorder.^{34} As will be discussed below, anion ordering seems to have an important impact on long-range ion diffusion in the I compound.
The 24g site represents an intermediate state used by the Li ions on 48h to diffuse through. As a result of these two lithium positions three different jump processes can occur; they are explicitly shown in Fig. 1b. (i) A strongly localised process is given by 48h–24h–48h′ jumps. Intracage jumps are possible within the octahedral cage when the Li ions jump between the 48h sites: (ii) 48h–48h′′. (iii) Long-range ion dynamics needs 48h_{1}–48h_{2} intercage jump processes between two different cages 1 and 2. The latter jump process is expected to be strongly affected by the anion site disorder of X and S^{2−}. We expect that a high degree of substitutional disorder will have a significant impact on the associated jump rate, which characterises exchange processes between the cages.
Fig. 4 (a) ^{31}P MAS NMR spectra of Li_{6}PS_{5}I, Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl recorded at 202.4 MHz and a spinning frequency of 25 kHz. The spectra have been referenced to 85% H_{3}PO_{4} and were recorded at an ambient bearing gas temperature.^{31} (b) ^{6}Li MAS NMR spectra of Li_{6}PS_{5}Cl, Li_{6}PS_{5}Br and Li_{6}PS_{5}I recorded at 73.6 MHz and a spinning speed of 25 kHz. All spectra have been referenced to solid LiCH_{3}COO. Values given in ppm indicate the isotropic chemical shifts. |
The ^{31}P MAS NMR spectrum of Li_{6}PS_{5}I is in stark contrast to the situation that is met for Li_{6}PS_{5}Cl, for which we expect the influence of strong S/Cl disorder on the line shape, see above. Indeed, its very broad line, which can hardly be resolved, reveals a wide distribution of chemical shifts and strong spin–spin interactions. Chemical shifts of lines constituting the overall line range from 85 to 81 ppm. Therefore, ^{31}P MAS NMR clearly reveals anion disorder in Li_{6}PS_{5}Cl. The ^{31}P MAS NMR line of Li_{6}PS_{5}Br takes an intermediate position. It is composed of narrower lines than those seen for Li_{6}PS_{5}Cl; nevertheless, line broadening, as compared to Li_{6}PS_{5}I, reveals disorder in the S/Br sublattice. At least the spectrum is composed of three distinct lines located at 93.9 ppm, 92.7 ppm and 91.1 ppm, respectively. The same increase in substitutional disorder might be seen if we consider the corresponding ^{127}I, ^{79}Br and ^{35}Cl MAS NMR spectra of the three compounds; the corresponding spectra of the three quadrupole nuclei are shown in the ESI,† Fig. S3. Clearly, because of the large quadrupole moment of I (|Q(^{127}I)| = 0.721b) the full spectrum including its spinning side bands is rather broad. The isotropic signal, however, is much smaller than that for Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br with |Q(^{35}Cl)| = 0.855b and |Q(^{79}Cl)| = 0.330b. Although exposed to less second order quadrupole effects, the line of Li_{6}PS_{5}Cl is as broad as that of the Br compound. Site disorder in Li_{6}PS_{5}Cl might explain this additional broadening.
The ordered anion framework seen for Li_{6}PS_{5}I by ^{31}P MAS NMR, synchrotron X-ray diffraction and neutron diffraction^{34} is fully consistent with results from ^{7}Li SLR NMR on the ion dynamics in the I compound. As we will show and discuss below, an almost symmetric diffusion-induced ^{7}Li NMR rate peak describes longitudinal spin–lattice relaxation in Li_{6}PS_{5}I; Li^{+} disorder on the split-site does not influence this symmetry. This observation mainly points to the absence of pronounced site disorder in the anion sublattice. Structural anion disorder is, thus, expected to have a significant impact on both the Li ion diffusion and ionic conductivity in Li_{6}PS_{5}X.
For the sake of completeness, we also recorded ^{6}Li (I = 1) MAS NMR spectra at an ambient bearing gas temperature, these spectra are shown in Fig. 4b. They have been referenced to LiCH_{3}COO. For Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl we observe a slightly asymmetric line with chemical shifts of δ_{iso} = 1.6 ppm and δ_{iso} = 1.3 ppm, respectively. The line of Li_{6}PS_{5}I turned out to be more symmetric in shape and is broader than the other lines. As Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl show rapid Li ion long-range translational dynamics their lines represent already coalesced spectra at the measurement temperature of 307 K, see the ESI.† This observation is in perfect agreement with ^{7}Li NMR motional line narrowing (see below), which has reached its extreme limit already well below ambient temperature.
In contrast, complete line narrowing of the static ^{7}Li NMR line of Li_{6}PS_{5}I is shifted toward much higher temperature, revealing significantly slower long-range ion dynamics (vide infra). In addition, the absence of effective Li ion dynamics in Li_{6}PS_{5}I, which would be able to also average (small) second order quadrupolar interactions that the spin-1 nucleus ^{6}Li is exposed to, might further increase the width of the NMR line of Li_{6}PS_{5}I as compared to the samples with X = Cl or Br. The increased anisotropy seen for Li_{6}PS_{5}Cl is in line with the degree of site disorder in Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br.
Interestingly, according to the halogen anion introduced into the argyrodite struture the isotropic chemical shift values δ_{iso} of the ^{6}Li MAS NMR spectra increase from Li_{6}PS_{5}Br to Li_{6}PS_{5}I and further to Li_{6}PS_{5}Cl. Thus, the paramagnetic component of the NMR line decreases; obviously, the different halogen anions change the electron density distribution in the direct neighborhood of the Li spins with Li_{6}PS_{5}Br having the highest density.
At low frequencies and high temperatures T the isotherms σ′(ν) reveal electrode polarization, which results from the accumulation of ions at the surface of the ion blocking electrodes applied.^{52–54} Towards higher frequencies or at sufficiently low T, distinct direct current (DC) plateaus show up, which reflect long-range ion transport. The corresponding σ_{DC} value is governed by both bulk and grain boundary contributions. For the bromide sample, Li_{6}PS_{5}Br, log_{10}(σ_{DC})T follows Arrhenius behaviour
σ_{DC}T = σ_{0}exp(−E_{a,DC}/(k_{B}T)) | (1) |
For comparison, in Fig. 5b the temperature dependence of log_{10}(σ_{DC})T of the other samples is presented. Li_{6}PS_{5}I and Li_{7}PS_{6} are characterised by the lowest conductivities and the highest activation energies E_{a,DC} of 0.47 eV and 0.37 eV. At approximately 250 K we notice a slight deviation from the Arrhenius line determining log_{10}(σ_{DC})T at higher T. This behaviour cannot be related to the phase transition of Li_{6}PS_{5}I;^{31,39,40,49} the I compound transforms into a low-T phase at T ≈ 160 K, see below. Such a slight deviation is also seen for Li_{6}PS_{5}Cl.
In Fig. 6a the changes in activation energies, ionic conductivities and Arrhenius pre-factors (σ_{0}, see eqn (1)) are illustrated. In all cases, the replacement of sulfur anions in Li_{7}PS_{6} by heavier halide anions leads to a decrease in activation energies. Except for Li_{6}PS_{5}Cl and Li_{6}PS_{5}Cl_{0.75}Br_{0.25} the Arrhenius pre-factor σ_{0} also decreases. This feature, which is seen when Li_{6}PS_{5}Br is compared with Li_{7}PS_{6} (or Li_{6}PS_{5}Cl), has been interpreted in terms of the Meyer–Neldel^{56} rule and related to a reduction in Debye frequencies for soft lattices created by substitution of Br and I for S.^{34} If we compare our results for Li_{6}PS_{5}Cl with respect to the conductivity behaviour of Li_{7}PS_{6} we witness both a decrease in E_{a,DC} from 0.536 eV to 0.396 eV and an increase in σ_{0} by one order of magnitude. This favorable combination of E_{a,DC} and σ_{0} results in a room temperature ionic conductivity of σ_{DC} = 3.8 mS cm^{−1}. For Li_{6}PS_{5}Br, with its more polarisable anions on the other hand, the low activation energy is compensated by a pre-factor two orders of magnitude lower than that for Li_{6}PS_{5}Cl. Considering high temperatures this feature causes σ_{DC} to adopt values lower than those of Li_{6}PS_{5}Cl.
Interestingly, the ionic conductivity of ordered Li_{6}PS_{5}I is even lower than that of the non-substituted compound Li_{7}PS_{6}; this decrease, despite the higher activation energy found for Li_{7}PS_{6}, can also be explained by the relatively low value for σ_{0}. The pre-factor is proportional to the attempt frequencies ν_{a}, the number of effective charge carriers n_{Li} and an entropy term.^{42} Besides the effect of ν_{a}, trapping effects in Li_{6}PS_{5}I, because of a very low intercage jump rate, might reduce N for this sample. If we assume similar ν_{a} values for Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br, entropy effects could serve as an explanation for the change in σ_{0}. Especially for the highly disordered Li_{6}PS_{5}Cl sample^{50} the migration entropy might take a decisive role in influencing the ion dynamics. For the ordered Li_{6}PS_{5}I compound this contribution might be absent.
When going from Li_{6}PS_{5}Br to Li_{6}PS_{5}I we see that, despite the introduction of polarisable iodine anions, the pre-factor remains almost the same, A = log(σ_{0}/(S cm^{−1} K)) = 4.9 for Li_{6}PS_{5}Br and A = 4.6 for Li_{6}PS_{5}I, but E_{a,DC} increases from 0.296 eV to 0.47 eV. From this point of view, although the lattice gets softer in the case of I, Kraft et al. mentioned^{34} that Meyer Neldel seems to be not fully applicable to describe the situation in Li_{6}PS_{5}I. One could have expected the more polarisable I anions to allow the Li ions to squeeze through smaller voids.^{50} Obviously, this concept does not work for Li_{6}PS_{5}Br and Li_{6}PS_{5}I, at least if we regard the (overall) averaged long-range ion transport process as probed by σ_{DC} measurements.
The ionic conductivities of the mixed compounds containing I anions, viz. Li_{6}PS_{5}Br_{0.5}I_{0.5} and Li_{6}PS_{5}Cl_{1/3}Br_{1/3}I_{1/3}, do not vary much. Obviously, the introduction of I plays the crucial role to lower σ_{DC}. The Cl anions do not affect the ionic transport in these high-entropy compounds in a positive way, at least for compounds containing two anions greatly differing in ionic radius (181 pm vs. 216 pm). To take a closer look at the exceptional behaviour of Li_{6}PS_{5}Cl with its high degree of anion disorder we tried to separate the bulk and grain boundary contributions of this sample. Moreover, we used complex resistivity measurements to further characterise the ion dynamics.
The σ′(ν) isotherms of Li_{6}PS_{5}Cl are shown in Fig. 7a. A careful look at the DC plateau region shows that it contains a small step. Thus, it is composed of actually two plateaus. Below 250 K, the corresponding activation energies turned out to be 0.310 eV and 0.305 eV, see Fig. 7b. The two-step behaviour of the σ′(ν) isotherms of Li_{6}PS_{5}Cl are also seen if we plot the real part ε′ of the complex permittivity vs. ν (Fig. 7b). We used the equation for a parallel-plate capacitor to estimate the corresponding capacitance C of the two steps:
(2) |
For comparison, the two contributions in the electrical relaxation of Li_{6}PS_{5}Cl are also seen in the loss factor tanδ = ε′′/ε′ when plotted as a function of frequency ν (see also Fig. 7b). The maxima in tanδ, showing up at 15.36 Hz (g.b.) and 4.21 kHz (bulk), correspond to the plateaus or inflection points of ε′ν, see also Fig. 8. According to = iωε_{0} = iωε_{0}ε′ + ωε_{0}ε′′ the maxima in tanδ produce two minima if the frequency dependence of the imaginary part σ′′(∝ε′) is analyzed (Fig. 8).
Fig. 8 Comparison of the frequency dependence of tanδ, ε′ and σ′′ of Li_{6}PS_{5}Cl measured at 173 K. Plateaus (or inflection points) in ε′ result in minima in σ′′ and maxima in tanδ, respectively. |
To further characterise ion hopping in Li_{6}PS_{5}X, especially if we consider samples with lower conductivity than Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br, we used the electric modulus representation^{58,59} to take a look at electrical relaxation frequencies τ_{M′′}^{−1} = ν_{max}. As an example, in Fig. 9 for Li_{6}PS_{5}Br_{0.5}I_{0.5} the imaginary part M′′ of the complex modulus is plotted vs. ν. Since the amplitude of M′′ is proportional to 1/C, bulk processes are mainly seen. For samples with very high conductivities, such as Li_{6}PS_{5}Cl, the peaks are shifted to too high frequencies and are, thus, no longer fully visible. Activation energies E_{a,M′′} from our M′′(ν) analysis are also included in Fig. 6. For comparison, the bulk activation energy deduced from Fig. 7 is included, too. They reveal that the activation energies, which we ascribe to bulk properties, do not change much if we consider compounds up to the composition Li_{6}PS_{5}Br_{0.5}I_{0.5}. This behaviour changes if we increase the I concentration. Importantly, for the compounds rich in I, especially if we consider Li_{6}PS_{5}I, the observation E_{a,M′′} < E_{a,DC} suggests that the charge carrier concentration increases with temperature. For Li_{6}PS_{5}I the difference E_{a,DC} − E_{a,M′′} turned out to be relatively high, viz. 0.121 eV (see the vertical arrow in Fig. 6).
Additionally, after we took a look at the overall and bulk ion dynamics in Li_{6}PS_{5}X we used complex resistivity measurements, = 1/ = /(iωε_{0}), carried out at constant frequency but varying temperature, to answer the question of how the bulk ion dynamics depends on the time-scale and length-scale that the technique applied is sensitive to. In Fig. 10a the temperature dependence of the real part ρ′ = M′′/ω of the complex resistivity is shown for Li_{6}PS_{5}Br. The peaks in Fig. 10a have been recorded for two different frequencies viz. 1 MHz and 10 MHz.
ρ′ is given by a Lorentzian-shaped function containing the electrical relaxation time τ_{ρ}:^{60}
(3) |
We cannot see any strong variation in E_{a,ρ} when comparing the response of Li_{6}PS_{5}Br_{0.5}I_{0.5} with that of Li_{6}PS_{5}Br. E_{a,ρ} values have also been included in Fig. 6; for the sake of clarity, they are separately shown in Fig. 10c. Obviously, a lower pre-factor of the underlying Arrhenius relation is responsible for this shift. The same effect is seen for Li_{6}PS_{5}I. Although the lattice is expected to be much softer in the case of Li_{6}PS_{5}I as compared to the situation in Li_{6}PS_{5}Cl, ρ(ω) tells us that the activation energies E_{a,ρ} of the two compounds are very similar (0.32 eV vs. 0.35 eV). However, the peak ρ′(ω) of Li_{6}PS_{5}I is clearly shifted toward higher T and shows up at 385 K; the origin of this difference has to be looked for in the bulk ion dynamics, g.b. resistances cannot explain it. As compared to the peak of Li_{6}PS_{5}Cl the peak of the I compound is symmetric in shape (β ≈ 1.95) and points to an ordered structure.
Obviously, as in the case of E_{a,DC} the reason for the lower conductivity and, thus, the slower bulk electrical relaxation in Li_{6}PS_{5}I (and in Li_{6}PS_{5}Br_{0.5}I_{0.5}) has been looked for in the prefactor of the underlying Arrhenius relationships. Most likely, the anion site disorder in Li_{6}PS_{5}Cl is responsible for this behaviour. Indeed, if we approximate the ρ′(ν) peaks with eqn (3) we can extract pre-factor τ_{0,ρ}^{−1} assuming τ_{ρ}^{−1} to be depending on temperature according to Arrhenius. τ_{0,ρ}^{−1} values are also included in Fig. 6; indeed, they clearly reveal lower pre-factors for Li_{6}PS_{5}I by approximately two orders of magnitude. If we assume that the attempt frequencies are similar for Li_{6}PS_{5}Cl and Li_{6}PS_{5}I a significant change in migration entropy, as mentioned above, might explain the prominent change seen. To check whether this phenomenon is also apparent in SLR NMR, which is in general sensitive to both short-range and long-range ion dynamics, we carried out ^{7}Li NMR measurements in both the laboratory and rotating-frame of reference.^{64,65} Line shape measurements carried out at temperatures down to the cryogenic region complement our investigation.
At higher T we expect spin–lattice relaxation to be increasingly induced by Li self-diffusion.^{61} Indeed, the rates sharply increase with temperature and characteristic diffusion-induced rate peaks show up. We recognise that R_{1} passes through well-defined maxima as has been found by some of us in an earlier study on the ion dynamics in Li_{6}PS_{5}Br.^{15} At the temperature T_{max}, where R_{1} reaches its maximum value, the motional correlation rate 1/τ_{c} is given by τ_{c}ω_{0} ≈ 1. 1/τ_{c} is identical within a factor of two to the Li jump rate 1/τ.^{61,65} The same holds for R_{1ρ}, for which the maximum condition is ω_{1}τ ≈ 0.5.^{61,65} As ω_{0} and ω_{1} differ by more than three orders of magnitude, the SLR NMR experiments, when taken together, are able to sense motional processes covering a large time scale.^{67,68}
We notice that the rate peak of Li_{6}PS_{5}Br shows up at T_{max} = 286 K. The peak is asymmetric and we used a Lorentzian-type (BPP-type)^{62,69} spectral density function J(ω_{0}) ∝ R_{1} of the form^{70}
(4) |
Here, the analysis of the rate peak of Li_{6}PS_{5}Br with eqn (4) yields an activation energy E_{a,NMR} of 0.213 eV, which is in perfect agreement with earlier findings for this compound.^{15,50} As mentioned above, the parameter β describes the deviation from symmetric behaviour (β = 2). β turned out to be β ≈ 1.5,^{65} which gives rise to a much lower activation energy E_{a,low} on the low-T side of the peak. In this regime, characterised by τ_{c}ω_{0} ≪ 1, short-range or localised ion dynamics in double-well potentials, including also highly correlated forward–backward jumps, are sensed by R_{1}.^{61} The activation energy of the low-T flank is given by 0.10 eV, which is in very good agreement with that seen by ρ′(ω). Localised intracage jump processes involving the 48h and 24g sites in Li_{6}PS_{5}Br could be mainly responsible for spin–lattice relaxation in this region, as has been precisely calculated (0.11 eV) by De Klerk et al.^{50} The higher activation energy E_{a,high}(=E_{a,NMR}) seen in the limit τ_{c}ω_{0} ≪ 1 might, however, additionally be affected by 48h–48h jumps. Usually in this regime, in which many jump processes occur during one Larmor precession of the spin, E_{a,NMR} should be comparable with E_{a,DC} (0.296 eV) or E_{a,ρ} (0.250 eV).^{61}E_{a,DC} could be influenced by g.b. effects; here, E_{a,NMR} < E_{a,ρ} shows that the number of jump events sensed by NMR, which should not necessarily include all types of jumps needed for long-range diffusion, is sufficient to generate a full R_{1} peak.
A very similar behaviour is found for Li_{6}PS_{5}Cl. The corresponding rate peak is shifted toward higher temperature revealing that faster ion dynamics is present in Li_{6}PS_{5}Br. In agreement with conductivity spectroscopy and our ρ′(ω) analysis, for Li_{6}PS_{5}Cl higher activation energies (E_{a,high} = 0.32 eV, E_{a,low} = 0.17 eV) than those seen for the Br analogue were found. Although E_{a,high} = 0.32 eV perfectly agrees with E_{a,ρ}, see Fig. 10, and calculated values,^{35} it is higher than expected from R_{1ρ} NMR in the limit τ_{c}ω_{1} ≪ 1 (0.248 eV). Furthermore, the shapes of the R_{1}(1/T) peaks of the two compounds are very similar. Obviously, similar (local) jump processes influence the NMR rates. In Table 1 an overview of the parameters is given obtained from the analysis with individual BBP-type spectral densities for each peak.
Li_{6}PS_{5}Cl^{a} | Li_{6}PS_{5}Br^{a} | Li_{6}PS_{5}I^{b} | |
---|---|---|---|
a The best R_{1ρ} fit is obtained if we replace ω_{1}/2π = 20 kHz by a slightly higher effective locking frequency ω_{eff} that also takes into account local magnetic fields. For Li_{6}PS_{5}Cl several runs of the global fit procedure yields ω_{eff} = 1.68 × ω_{1}; for Li_{6}PS_{5}Br ω_{eff} is given by ω_{eff} = 1.41 × ω_{1}. b Li_{6}PS_{5}I undergoes a phase transition at 160 K; thus, no β_{ρ} can be observed. The symmetric rate peak (β ≈ 2) agrees with the ordered anion sublattice as suggested by XRPD, see above. | |||
β | 1.53(2) | 1.48(2) | 1.92(1) |
E _{a} | 0.320(1) eV | 0.213(1) eV | 0.198(1) eV |
E _{a,low} | 0.170(4) eV | 0.102(5) eV | 0.182(5) eV |
τ _{0} | 1.3(5) × 10^{−14} s | 2.3(5) × 10^{−13} s | 8.2(5) × 10^{−13} s |
β _{ ρ } | 1.54 | 1.41 | — |
E _{a,ρ} | 0.248(9) eV | 0.201(9) eV | 0.137(3) eV |
E _{a,low} | 0.134(1) eV | 0.083(1) eV | — |
τ _{0,ρ} | 9.5(5) × 10^{−13} s | 3.3(1) × 10^{−12} s | — |
Exactly the same trend is seen when we look at the R_{1ρ}(1/T) peaks of the two compounds, which are shown in Fig. 11b. These have been, independently of the R_{1} experiments, also parameterised with Lorentzian-shaped spectral density functions. The activation energies of the peak of Li_{6}PS_{5}Br (0.201 eV, 0.083 eV) match with those reported earlier^{15} also including calculated values.^{50} They also agree with those from the R_{1} peak. Here, R_{1ρ}(1/T) of Li_{6}PS_{5}Br turned out to be much smaller that that reported by Yu et al.;^{77} most likely, the difference has to be looked for in the preparation conditions such as milling steps and annealing procedures.
In Fig. 12 so-called joint fits^{65} of the two types of NMR experiments, R_{1} and R_{1ρ}, are shown. For the joint fits the parameters E_{a,NMR} and τ_{0}^{−1}, which is the pre-factor of the Arrhenius equation of τ_{c}^{−1}, have been linked to each other. The best fit is obtained if we use two independent values for β and if we replace ω_{1} by effective frequencies ω_{1,eff}, also taking into account local magnetic fields.^{73} To estimate the latter, we did not fix ω_{1} and looked at how the quality of the fit changes if it is a free parameter. Only a slight change of ω_{1} was necessary giving rise to ω_{eff} ≈ 1.5 × ω_{1}. A successful joint fit shows that the same overall diffusion processes govern the peaks R_{1} and R_{1ρ}. The results of our joint fits are shown in Table 2. The activation energies E_{a,NMR} (=E_{a,high}) obtained fulfill the relationship
E_{a,low} = (β_{(ρ)} − 1)E_{a,high}. | (5) |
Fig. 12 Arrhenius plot of the ^{7}Li NMR relaxation rates R_{1} and R_{1ρ} of (a) Li_{6}PS_{5}Cl and (b) Li_{6}PS_{5}Br measured in the laboratory frame of reference (116 MHz) and in the rotating frame of reference (20 kHz, nominal locking frequency). The solid lines represent the global fit based on a modified BPP model that takes into account peak asymmetries by the parameter β_{(ρ)}. The fastest Li diffusion is found for Li_{6}PS_{5}Br with the peak maxima showing up at 286 K and 168 K, respectively. The second R_{1ρ} relaxation process, most likely also influenced by spin–spin relaxation to a certain degree, is indicated, too. As mentioned before (see Fig. 11), the open symbols represent rates that were measured with an NMR probe designed for experiments at cryogenic temperatures; the filled symbols show rates which were acquired with the standard probe. |
E _{a,NMR} | C | β _{(ρ)} | τ _{0} | |
---|---|---|---|---|
a The best global fits are obtained if we replace ω_{1} by the effective locking frequencies ω_{eff} = 1.68 × ω_{1} (Li_{6}PS_{5}Cl) and ω_{eff} = 1.41 × ω_{1} (Li_{6}PS_{5}Br). The coupling constant C, which is the amplitude of the rate peak, J(ω_{0}) = Cτ_{c}/(1 + (ωτ_{c})^{β}), turns out to be on the order of 10^{9} to 10^{10} s^{−2}. | ||||
Li_{6}PS_{5}Cl | ||||
R _{1} | 0.273(5) eV | 3.5(5) × 10^{9} s^{−2} | 1.60(1) | 4.9(4) × 10^{−14} s |
R _{1ρ} | 3.1(5) × 10^{10} s^{−2} | 1.47(1) | ||
Li_{6}PS_{5}Br | ||||
R _{1} | 0.218(1) eV | 5.6(7) × 10^{9} s^{−2} | 1.48(1) | 2.0(2) × 10^{−13} s |
R _{1ρ} | 3.3(9) × 10^{10} s^{−2} | 1.47(1) | ||
Li_{6}PS_{5}I^{a} | ||||
R _{1} | 0.198(1) eV | 1.1(4) × 10^{10} s^{−2} | 1.92(1) | 8.2(9) ×10^{−13} s |
R _{1ρ} | 0.137(3) eV | — | — | — |
Considering the overall constants C obtained from the joint fits, they reveal that both dipolar and electric quadrupolar interactions govern the diffusion-induced SLR NMR rates. Estimating C_{dipolar} with the help of the rigid lattice line width for Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl would underestimate C to be on the order of 10^{10} s^{−2}.^{46} As we will discuss below, for Li_{6}PS_{5}Br quadrupolar interactions have been identified to greatly influence longitudinal relaxation.^{15}
Interestingly, a very close look at the R_{1ρ} magnetization transients revealed that they contain a minor magnetization component that leads to activation energies of 0.484 eV (Li_{6}PS_{5}Cl) and 0.298 eV (Li_{6}PS_{5}Br); the corresponding R_{1ρ} rates are included in Fig. 12. We cannot exclude that these rates are to a certain degree also influenced by spin–spin relaxation. Obviously, the rates sense long-range Li jump processes characterised by higher activation energies. The intercage jumps might contribute to this type of magnetic relaxation, too.
For X = Cl and Br, if we consider activation energies, T_{max}, β and β_{ρ}, the ^{7}Li SLR NMR data reveal the same trend for the ion dynamics as seen by the conductivity and ρ′(ω). Li_{6}PS_{5}Br shows fast ion dynamics and the lowest activation energies. For Li_{6}PS_{5}Cl we witness slightly higher activation energies; simultaneously, the peaks shift toward higher temperature. For Li_{6}PS_{5}I, on the other hand, significant differences show up (see the next section).
The NMR rate peaks of Li_{6}PS_{5}I, remembering its low σ_{DC} conductivity of 10^{−6} S cm^{−1} at ambient temperature, are expected to show up at temperatures much higher than 320 K. We would expect much higher activation energies, at least similar to or higher than those found for Li_{6}PS_{5}Cl. Surprisingly, the R_{1} rate peak of Li_{6}PS_{5}I shows up at a temperature highly comparable to that of Li_{6}PS_{5}Cl viz. at T_{max} = 329 K, see Fig. 11a. In stark contrast to our expectation, NMR does not point to slower Li^{+} diffusion. Hence, in Li_{6}PS_{5}I the same fast (local) jump processes are present as in their parent compounds.^{35,40,50} Here, we anticipate that the Li ions in Li_{6}PS_{5}I, with its ordered anion sublattice, have access to the same rapid intracage jump processes. According to NMR the barriers have to be characterised by an activation energy of 0.2 eV. Obviously, only intracage jump processes in Li_{6}PS_{5}I are sufficient to produce a full R_{1}(1/T) peak, see the discussion above.
As expected for a material with an ordered anion lattice such as Li_{6}PS_{5}I and without severe influences of motional correlation effects, an almost symmetric rate peak R_{1}(1/T) (β = 1.92(≈β′)) is obtained for Li_{6}PS_{5}I. The same shape was seen in ρ′(ω). Interestingly, an even lower activation energy determines R_{1ρ}(1/T), see Fig. 11c: in the limit τ_{c}ω_{1} ≪ 1 the rates follow an Arrhenius line whose activation energy E_{a,high} is given by only 0.137 eV. This value is remarkably similar to a cage-like local pathway studied by Rao and Adams (0.15 eV) who used a bond valence approach to investigate the Li ion dynamics in Li_{6}PS_{5}I. Our experimental value also coincides with that of Pecher et al.^{40} (0.14 eV) who studied the local ion dynamics by molecular dynamics simulations.
The amplitude (R_{1})_{max} of the R_{1}(1/T) rate of Li_{6}PS_{5}I turned out to be higher that that of the peaks belonging to Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br. This could be because of stronger heteronuclear Li–X interactions and the fact that Li ions on the distorted 24g sites give rise to stronger dipolar and electric quadrupolar couplings constants. These are lowest for the Li_{6}PS_{5}Cl compound. In general, (R_{1})_{max} is proportional to the square of the magnetic dipolar or electric quadrupolar coupling constant.^{75,78,79} If we use an electric quadrupole constant of at least 50 kHz, as has been estimated for Li_{7}PS_{6},^{46} we obtain (R_{1})_{max} ≈ 2 s^{−1} yielding log_{10}((R_{1})_{max}/s) ≈ 0.3, showing that for Li_{6}PS_{5}Cl both dipolar and electric quadrupolar interactions play a role in determining the relaxation mechanism. For Li_{6}PS_{5}Br we showed via comparative ^{7}Li and ^{6}Li SLR NMR measurements that quadrupole fluctuations greatly affect the overall spin–lattice relaxation.^{15}
Coming back to the temperature dependence of the SLR NMR rates of Li_{6}PS_{5}I, we see that at higher temperatures than ambient the rates R_{1ρ} are increasingly governed by R_{1}, which produces an apparent maximum at ca. 330 K. Unfortunately, the peak maximum of R_{1ρ}(1/T) cannot be detected as Li_{6}PS_{5}I undergoes a phase transition at 160 K;^{31,39,40,49} see the abrupt change in R_{1ρ} in Fig. 11c. The ion dynamics in the low-T (LT) phase is characterised by a much higher activation energy of 0.44 eV. Below 160 K the rates R_{1} are no longer induced by diffusive Li jump processes.
To explain the evident differences between the results from conductivity measurements and nuclear spin relaxation in Li_{6}PS_{5}I we anticipate that the fast diffusion Li processes in the I compound are not interconnected to give rise to long-range, through-going ion transport. As has been pointed out in detail by Wagemaker and co-workers, and first mentioned by Rao and Adams,^{35} the jump probability between two rings in Li_{6}PS_{5}I is very low.^{50} Intercage jump processes with higher activation energies lead to poor ion transport in Li_{6}PS_{5}I. On a shorter length scale, however, the same structural motifs in Li_{6}PS_{5}X cause very similar NMR rate peaks. Anion disorder, taking advantage of anions with radii that do not differ much to that of S^{2−}, switches on the intercage exchange processes, making Li_{6}PS_{5}Br a fast ion conductor. This design principle causes the overall activation energy to increase in the following order: Li_{6}PS_{5}Br < Li_{6}PS_{5}Cl < Li_{6}PS_{5}I.
To verify whether some intragrain translational processes needed for long-range diffusion in Li_{6}PS_{5}I are indeed missing or governed by higher activation energies we carefully looked at the R_{1ρ} transients and recorded ^{7}Li NMR line shapes over a broad temperature range (Fig. 13). Interestingly, the above-mentioned slow magnetization component seen for Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br (Fig. 12) is missing for the I compound. Furthermore, the intercage jumps are expected to contribute to the averaging of homonuclear (Li–Li) dipole–dipole interactions. If missing at low temperatures, for Li_{6}PS_{5}I the change in line width Δν should occur in two steps. Full averaging is expected at temperatures for which the intercage exchange rate reaches sufficiently high values on the order of some kHz. Indeed, whereas for Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl the ^{7}Li NMR line width continuously decreases with temperature revealing typical narrowing curves (Fig. 13a), that of Li_{6}PS_{5}I is different.
Fig. 13 (a) Temperature dependence of the ^{7}Li NMR line widths (FWHM = full width at half maximum) of the aryrodites Li_{6}PS_{5}X (X: Cl, Br and I). Some of the data points referring to Li_{6}PS_{5}Br and measured at temperatures lower than 80 K were taken from Epp et al.;^{15} they perfectly complement the results of the sample studied here. Line widths Δν_{r}(T) of Li_{6}PS_{5}Br start to narrow at temperatures as low as 86 K indicating ultrafast hopping processes that average homonuclear dipole–dipole interactions being responsible for the rigid lattice line width Δν_{r} ≈ 6.1 kHz. Li diffusion in the compound with Cl is slightly slower as the onset of line narrowing is shifted toward higher T. The line of Δν_{r}(T) of Li_{6}PS_{5}I narrows in steps of two. At T < 160 kHz it indicates dipole averaging in the low-T phase. Because inter-ring jump processes are absent, the line does not reach its extreme value at a temperature comparable to that of Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl. Higher temperatures are necessary to obtain a fully narrowed ^{7}NMR line. Lines represent fits according to the models of Hendrickson and Bray (dashed lines) and Abragam (solid lines). (b) ^{7}NMR line shapes of Li_{6}PS_{5}Br recorded at the temperatures indicated; the line transforms from a Gaussian shape at low T to a Lorentzian one at elevated temperatures. |
For Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl the extreme narrowing regime is already reached at 250 K (ν_{∞} ≈ 300 Hz); the final line width is caused by external field inhomogeneities. In contrast to the situation observed for Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl, at 250 K the ^{7}Li NMR line of Li_{6}PS_{5}I is still relatively broad and shows a width of ν = 2.2 kHz. At 350 K it finally reaches ν_{∞}. Thus, for Li_{6}PS_{5}I full averaging of the NMR line width is shifted by ΔT = 100 K toward higher T. Most likely, the latter step corresponds to intercage ion dynamics which are governed by a larger activation barrier. Hence, it is clear from NMR motional narrowing data that in Li_{6}PS_{5}I a much slower process kicks in only at higher T. The fact that Δν of Li_{6}PS_{5}I and Li_{6}PS_{5}Cl is very similar, at least at temperatures just above the phase transition, shows that mainly fast intracage jump processes are responsible for the narrowing seen in this T range. Differences show up, however, if we consider the rotating-frame spin–lattice relaxation rates, see above. The activation energies of the two samples differ by ca. 0.11 eV. Interestingly, the R_{1ρ}(1/T) peak of Li_{6}PS_{5}I is expected at a similar temperature to that of Li_{6}PS_{5}Br (0.20 eV), again pointing to very fast intracage jump processes in the iodide compound. Most likely, so-called doublet-jump processes^{50} have to be considered to explain this feature, see below.
Unfortunately, we cannot probe the rigid-lattice line width ν_{0} of the Li_{6}PS_{5}I compound because of the phase transition that occurs at 160 K.^{31,39,40,49} For both Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl the value of ν_{0} is very similar; ν_{0} turned out to be ca. 6.1 kHz. Therefore, the magnetic dipolar Li–Li interactions, which are determined by the Li–Li distances, are almost the same in the two compounds. The inflexion point of the motional narrowing curve of Li_{6}PS_{5}Br is located at T = 150 K (Fig. 13a). R_{1ρ}(1/T) tells us that at this temperature the motional correlation rate τ_{c}^{−1} (≈2ω_{1}) should be on the order of 2.51 × 10^{5} s^{−1}. This value clearly exceeds ν_{0} and causes full averaging of the spin–spin interactions in this T range. The changes of the corresponding ^{7}Li NMR lines are shown in Fig. 13b. Li_{6}PS_{5}Br shows one of the lowest onset temperatures seen by NMR motional line narrowing; already at 86 K the shape of the NMR line starts to change as a result of rapid Li^{+} self-diffusion.^{15} This observation excellently agrees with results from density functional theory molecular dynamics simulations.^{50}
Extracting quantitative information from NMR motional line narrowing is always fraught with difficulties.^{80} Abragam^{81} and Hendrickson and Bray^{82,83} introduced models to use the change in line widths to deduce activation energies for the hopping process behind. There is still debate over whether these models are accurate enough for this purpose. Hence, we should not overinterpret the results but regard them as estimates. The dashed lines in Fig. 13 refer to the following relationships^{82}
(6) |
(7) |
E _{a,A} (eV) | E _{a,HB} (eV) | E _{a} (low-T flank) (eV) | |
---|---|---|---|
Li_{6}PS_{5}Cl | 0.11(1) | 0.18(2) | 0.17(4) |
Li_{6}PS_{5}Br | 0.06(1) | 0.09(1) | 0.10(4) |
Li_{6}PS_{5}I | 0.23(1) | 0.38(2) | 0.18(5) |
Finally, we will quantitatively compare jump rates τ^{−1} from NMR with electrical relaxation rates from impedance and conductivity spectroscopy. Using the Nernst–Einstein equation^{84} we can convert σ′ into diffusion coefficients and, thus, jump rates. The easiest way, being also independent of any relaxation model, is to determine jump rates τ^{−1} at the temperatures where the NMR relaxation rate peaks show up. With the conditions τ_{c}ω_{1(ρ)} = 1(0.5), valid at T_{max}, and the jump distances obtained from XRD we estimated diffusion coefficients according to the Einstein–Smoluchowski relation^{84} for 3D diffusion
(8) |
/Å | T _{max}(R_{1})/K | D _{NMR}/cm^{2} s^{−1} | n _{Li}/m^{−3} | |
---|---|---|---|---|
For we used the 48h–48h distance, which is the longest jump distance in Li_{6}PS_{5}Br, when we consider cation hopping processes near the regularly occupied Li sites. | ||||
Li_{6}PS_{5}Cl | 2.34 | 316 | 6.7 × 10^{−8} | 6.3 × 10^{27} |
Li_{6}PS_{5}Br | 2.41 | 286 | 7.1 × 10^{−8} | 6.0 × 10^{27} |
Li_{6}PS_{5}I | 2.66 | 329 | 8.6 × 10^{−8} | 5.8 × 10^{27} |
Using structural data from our Rietveld analysis we also estimated the charge carrier density n_{Li} in Li_{6}PS_{5}Br. n_{Li}, which is on the order of 10^{27} m^{−3}, is needed to convert D_{NMR} into conductivities expected at or near T_{max}. Disregarding any correlation effects for our estimation of σ_{NMR}, the following relation holds^{11}
(9) |
σ _{NMR}/mS cm^{−1} | σ _{DC}/mS cm^{−1} | |
---|---|---|
Li_{6}PS_{5}Cl | 2.46 (316 K) | 9.00 (313 K) |
Li_{6}PS_{5}Br | 2.78 (286 K) | 2.20 (293 K) |
Li_{6}PS_{5}I | 2.80 (329 K) | 8.60 × 10^{−3} (333 K) |
While good agreement is seen for σ_{NMR} and σ_{DC} of Li_{6}PS_{5}Br, we have to notice a somewhat larger difference for the Cl compound. In general, conductivities indirectly deduced from NMR should be regarded as estimates because relaxation NMR and conductivity spectroscopy sense different kinds of motional correlation functions. Agreement over a large temperature is only obtained if both techniques are sensitive to the same diffusion process and the same correlation function.^{51}
As expected, for Li_{6}PS_{5}I the values listed in Table 5 differ by three orders of magnitude. Whereas σ_{DC} is mainly governed by successful jump processes resulting in long-range ion transport, σ_{NMR} reflects short-range ion dynamics. The difference can easily be explained by the fact that in Li_{6}PS_{5}I the intercage jump processes are less probable; these processes hinder the ions in long-range diffusion. Note that the comparison in Table 5 only refers to T ≈ T_{max} at which the NMR peaks show up. Comparing dynamic parameters over the full temperature range will show deviations between nuclear spin relaxation and conductivity spectroscopy as the techniques deliver different activation energies in the present case. Such a comparison is presented in Fig. 14. Squares represent jump rates obtained at the NMR rate peaks; at T → 0 we have indicated the pre-factors of the Arrhenius lines.
Fig. 14 Arrhenius plot of the jump rates deduced from the ^{7}Li NMR rate peaks shown in Fig. 11c, 12a and (b). (a) Squares represent either pre-factors or jump rates at the maxima of the peaks. Solid lines represent the Arrhenius lines according to the joint fits. Dashed lines indicate the position of τ_{σ}^{−1}, i.e., estimated from σ_{DC}. (b) The same data points as in (a) but with τ_{DC}^{−1} included as data points. The lines referring to Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br have been shifted by a factor of +4 (+2) (on the log scale) to illustrate the kink seen in τ_{DC}^{−1}(1/T) for Li_{6}PS_{5}Cl. It is absent for Li_{6}PS_{5}Br. Electrical relaxation rates τ_{ρ}^{−1} from our ρ′(1/T) analyses have been included as well. The vertical arrow drawn with a dashed line indicates the large difference in jump rates seen for Li_{6}PS_{5}I. The arrows near the ordinate illustrate the change in pre-factors. See the text for further explanation. |
In Fig. 14aτ_{NMR}^{−1} as deduced from the joint fits of Li_{6}PS_{5}Cl and Li_{6}PS_{5}Br are shown. The line belonging to Li_{6}PS_{5}I refers to the R_{1} rates only. First of all, we notice a correlation between E_{a,NMR} and the pre-factor τ_{0}^{−1} (see also Table 2). The decrease in E_{a,NMR}, which changes from 0.273 eV to 0.218 eV and further to 0.198 eV (R_{1} of Li_{6}PS_{5}I), is accompanied by a decrease in τ_{0}^{−1}. The attempt frequency τ_{0}^{−1} decreases by a factor of 80 when going from Li_{6}PS_{5}Cl to Li_{6}PS_{5}I. This small variation either reflects a decrease in phonon frequencies or a change in migration entropy ΔS of the ions as τ_{0}^{−1} = ν_{p,0}exp(ΔS/k_{B}). Phonon frequencies, to which ν_{p,0} is proportional, usually take values on the order of 10^{12} to 10^{15} s^{−1}. As discussed above, for Li_{6}PS_{5}I a decrease in migration entropy is reasonable because of the ordered anion sublattice. The decrease in E_{a,NMR} reflects the increase in anion polarizability (I > Br > Cl), which should, apart from effects due to disorder, also lead to lower hopping barriers. This trend for E_{a,NMR} is at least valid for the elementary Li^{+} jumps seen by NMR. If we include τ_{σ}^{−1} values obtained by transforming σ_{DC} into hopping rates after eqn (9), we notice that good agreement is only seen at temperatures around T_{max}, see Fig. 14a and b. Clearly, the fact that E_{a,NMR} ≠ E_{a,DC} leads to deviations at temperatures T ≶ T_{max}. For comparison, in Fig. 14b hopping rates from ρ′(T) have also been included.
The pre-factors of the Arrhenius lines referring to τ_{σ}^{−1} estimated from σ_{DC} also vary with X. While we cannot see any large difference between τ_{0,σ}^{−1} from Li_{6}PS_{5}I and Li_{6}PS_{5}Br as mentioned above, we see that τ_{0,σ}^{−1} of the Cl-compound is much higher than expected. This increase is caused by a change in slope of σ_{DC} at a temperature slightly below 400 K; it is also seen for Li_{6}PS_{5}I. Below 400 K the ionic conductivity σ_{DC} (as well as τ_{σ}^{−1}) follows an Arrhenius law with an activation energy very similar to that of Li_{6}PS_{5}Br (0.273 eV). As σ_{DC} of Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl almost coincide in this T range, also the corresponding pre-factors τ_{0,σ}^{−1} would be the same. Here, the fact that τ_{0,σ}^{−1} of Li_{6}PS_{5}Cl is larger than that of Li_{6}PS_{5}Br originates the kink in the Arrhenius line causing σ_{DC}T(1/T) to be characterised by an activation energy (0.396 eV) much larger than that in the low-T range. Obviously, another Li^{+} diffusion process, being characterised by an activation energy larger than 0.3 eV, contributes to τ_{σ}^{−1} of Li_{6}PS_{5}Cl.
Such an additional process, which we assume could stem from an ordered Li sublattice in Li_{6}PS_{5}Cl, has been seen by R_{1ρ}, see Fig. 12. The R_{1ρ} magnetization transients of Li_{6}PS_{5}Cl are characterised by a second decay process that is thermally activated by 0.48 eV. The same feature is also seen for Li_{6}PS_{5}Br with both a disordered anion structure and Li^{+} sublattice. The corresponding activation energy of 0.30 eV (see Fig. 12) is, however, very similar to that of σ_{DC}T(1/T). This similarity thus gives rise to no increase in σ_{DC}T above T ≈ 400 K.
The pre-factors seen by conductivity spectroscopy, especially those of Li_{6}PS_{5}Cl, clearly represent mean values. In materials with several diffusion processes taking place in parallel, each process needs to be characterised by its individual pre-factor. In the case of ^{7}Li NMR, which is primarily sensitive to the elementary jump processes between 24g and 48h, we see a slight increase in the pre-factor when going from X = I to X = Cl (see above). The change in τ_{0}^{−1} is, however, small as compared to the drastic decrease in σ_{DC}T(1/T) seen for Li_{6}PS_{5}I. Hence, the increase in overall migration enthalpy for Li_{6}PS_{5}I, for which the intercage diffusion process is characterised by a high hopping barrier, turned out to be mainly responsible for its poor ionic conductivity. On shorter length scales, Li ion hopping in Li_{6}PS_{5}I, with its ordered anion sublattice, is as fast as in Li_{6}PS_{5}Br. This result is in perfect agreement with the pioneering calculations by Rao and Adams as well as the in-depth study presented by De Klerk et al.^{50} They already pointed out, as mentioned above, that for Li_{6}PS_{5}I the intercage jump processes are interrupted and fast localized motions with very low activation energies are present. By ^{7}Li NMR relaxometry their observation is now fully supported. The calculations by Pecher et al. point in the same direction.^{40} As an example, De Klerk et al. report an activation energy as low as 0.05 eV for the doublet jump process in Li_{6}PS_{5}I.^{50}
Besides these peculiarities for Li_{6}PS_{5}Cl and Li_{6}PS_{5}I, the complex ion dynamics in Li-bearing argyrodites with anion disorder is reflected in the distribution of activation energies. Table 6 summarises the activation energies obtained by the different methods applied. Depending on the length-scale and time-scale to which the specific method is sensitive quite different activation energies can be probed. Such a result is typical for ionic conductors whose charge carriers are exposed to an irregularly shaped energy landscape that gives rise to highly correlated forward and backward motions or fast (local) dynamics on the angstrom length scale.
Li_{6}PS_{5}Cl (eV) | Li_{6}PS_{5}Br (eV) | Li_{6}PS_{5}I (eV) | |
---|---|---|---|
a The values referring either to the high-T or to the low-T flank were obtained by a line fit. b Results from the global fit analysis are obtained by linking E_{a} and τ_{0}^{−1} of the individual relaxation rate peaks. Values in brackets refer to the low-T flanks of the global fit. | |||
σ _{DC} | 0.396(3) | 0.296(2) | 0.470(5) |
ρ′, high-T flank^{a} | 0.321(3) | 0.250(3) | 0.347(7) |
ρ′, low-T flank^{a} | 0.153(2) | 0.115(9) | 0.305(2) |
M′′ | — | — | 0.349(5) |
R _{1}, high-T flank^{a} | 0.320(1) | 0.213(1) | 0.198(1) |
R _{1}, low-T flank^{a} | 0.170(4) | 0.102(4) | 0.182(5) |
R _{1ρ}, high-T flank^{a} | 0.248(9) | 0.201(9) | 0.137(3) |
R _{1ρ}, low-T flank^{a} | 0.134(1) | 0.083(1) | — |
R _{1} (global fit)^{b} | 0.273(5) | 0.218(1) | — |
(0.165(8)) | (0.105(4)) | — | |
R _{1ρ} (global fit)^{b} | 0.273(5) | 0.218(1) | — |
(0.129(7)) | (0.081(4)) | — |
Anion disorder in Li_{6}PS_{5}Br and Li_{6}PS_{5}Cl, as verified by X-ray powder diffraction measurements, boosts the ion dynamics as compared to the parent compound Li_{7}PS_{6}. The halogen anions are clearly distributed over the crystallographic sites 4a and 4d. Li_{6}PS_{5}Br reveals the lowest activation energies and the bulk ion conductivities follow a single Arrhenius line. Deviations from this behaviour are seen for Li_{6}PS_{5}Cl and Li_{6}PS_{5}I. For Li_{6}PS_{5}Cl an additional dynamic process shows up at higher T, which is also seen in ^{7}Li spin-lock NMR spectroscopy. This process leads to a relatively high mean activation energy in σ_{DC} characterising the overall ion transport.
Most importantly, also for the poor ionic conductor Li_{6}PS_{5}I our NMR relaxation measurements reveal very fast Li ion dynamics on a local to medium-range length scale. Obviously, the ions have access to the same, rapid exchange processes as in Li_{6}PS_{5}Br but long-range transport is switched off. In the spirit of De Klerk and co-workers we think that the ordered anion sublattice, combined with the larger lattice constant, is responsible for this peculiarity. It leads to a heterogeneous potential landscape with low and high hopping barriers. High activation barriers seem to characterize the intercage jumps; these jump processes are, however, necessary to transport the ions over long distances.
Footnote |
† Electronic supplementary information (ESI) available: Rietveld refinements and structural data, further NMR data. See DOI: 10.1039/c9cp00664h |
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