Estimation of the concentration and mobility of mobile Li⁺ in the cubic garnet-type Li₇La₃Zr₂O₁₂

Abstract

Li₇La₃Zr₂O₁₂ (LLZ) lithium ion conductors with the garnet-like structure are promising candidates for applications in all solid-state lithium ion batteries. Due to the complexity of the structure and the distribution of Li⁺, it was difficult to get information on the true concentration of mobile Li⁺, n_c, and their mobility. In this report, we estimate for the first time the values of n_c from the analysis of the conductivity spectra at different temperatures. We found that only a small fraction of Li⁺ contributes to the conduction process. n_c is found to be independent of temperature with an average value of 3.17 × 10²¹ cm⁻³, which represents 12.3% only of the total Li⁺ content in LLZ garnets. Comparing the conduction parameters of LLZ with Li₆BaLa₂Ta₂O₁₂ (LBLT) and Li₅La₃Ta₂O₁₂ (LLT) garnets indicates that the mobility of Li⁺ plays a prominent role in the conductivity enhancement in LLZ garnet materials. The diffusion coefficient of LLZ at room temperature has a value of 1.33 × 10⁻⁸ cm² s⁻¹, which is comparable with other fast Li⁺ conductors.

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1. Introduction

Solid lithium ion conductors with high ionic conductivity are promising candidates for applications in different electrochemical devices. Among these materials, Li⁺ conductors with the garnet-like structure of the formula Li₅La₃M₂O₁₂ (M = Ta, Nb) have good ionic conductivity in the range of 10⁻⁶ S cm⁻¹ at room temperature with a small contribution from the grain boundaries and negligible electronic conductivity.^1,2 These garnet materials are also stable against reactions with metallic lithium electrodes and are stable in air at ambient and high temperatures.^1–5 These features make lithium conducting garnets favored solid electrolytes in all solid-state lithium ion batteries.^1–5 Extensive work has been devoted to optimize the ionic conductivity of garnet materials, which could be achieved by chemical substitutions and structural modifications.^4–21 Chemical substitutions of divalent or trivalent cations in the La or M sites, respectively in Li₅La₃M₂O₁₂ garnets lead to increasing the lithium content, associated with considerable enhancement of the ionic conductivity up to 10⁻⁴ S cm⁻¹ at room temperature.^4–21 The most extensively studied members of the lithium conducting garnets are the cubic Li₇La₃Zr₂O₁₂ (LLZ) based materials due to their high ionic conductivity >10⁻⁴ S cm⁻¹ at RT.^4,12–21 Further enhancement of the conductivity up to 10⁻³ S cm⁻¹ was achieved in different chemically substituted LLZ garnets.^12–21

In order to develop new fast Li⁺ conducting garnets it is essential to understand the structure-property relationship, the lithium distribution and the lithium conduction mechanism. In garnet materials Li⁺ could occupy either the 24d tetrahedral sites or the 48g/96h octahedral sites.^22–29 However, it is found that the occupancy of the tetrahedral or octahedral sites depends on Li⁺ content in the garnet phases.^23–29 For the original garnet structure, such as Li₃Nd₃Te₂O₁₂, Li⁺ occupy all the available 24 tetrahedral sites, with the octahedral sites left empty. Li⁺ in this garnet phase are immobile and the Li₃Nd₃Te₂O₁₂ phase has un-measurable conductivity at RT.^25,26 With increasing Li⁺ content per unit formula re-distribution of Li⁺ occurs, where the occupancy of the tetrahedral sites decreases associated with increased occupancy of the octahedral sites. The occupancy of the 24d sites is 80%, 67% and 56%, whereas that of the octahedral sites is 43%, 64% and ∼90% for Li⁺ content per unit formula of 5(Li₅La₃Ta₂O₁₂), 6(Li₆BaLa₂Ta₂O₁₂) and 7(Li₇La₃Zr₂O₁₂), respectively.^23–29 The ionic conductivity values of LLT, LBLT and LLZ phases are 10⁻⁶, 4 × 10⁻⁵ and 3 × 10⁻⁴ S cm⁻¹, respectively at RT.^1–5

The above results may suggest that the ionic conduction in garnet materials is associated with the transport of Li⁺ occupying the octahedral sites only.^22,23 However, recent experimental and computational studies concluded that Li⁺ conduction involves both tetrahedral and octahedral sites.^26–29 The dc conductivity could be described by the following relation;

σ_dc = en_cμ (1)

where e is the electronic charge, n_c is the concentration of mobile charge carriers and μ is their mobility. Therefore, the primary factors that control the ionic conduction process are the concentration and mobility of Li⁺. However, the complex nature of the structure and the local cations arrangement in the garnet materials make it difficult to get information on the mobility and concentration of Li⁺. In the present work we aim to estimate the concentration of mobile Li⁺ at various temperatures that participate in the conduction process and their mobility by the analysis of the conductivity spectra of LLZ lithium conducting garnets.

2. Experimental

Li₇La₃Zr₂O₁₂ (LLZ) powder was prepared by a combination of mechanical milling and solid state reaction techniques. Stoichiometric amounts of Li₂CO₃ (10 wt% excess of Li₂CO₃ was added to compensate for lithium loss at high temperatures), ZrO₂ and La₂O₃ (dried at 900 °C overnight) were mixed together and calcinated in three different temperatures of 700, 900 and 1000 °C for a period of 12 h for each step. Before and after each calcination step the powder was ball milled in 2-propanole for 12 h in tungsten carbide media with a rotation speed of 300 rpm, except for the final milling step where the rotation was 500 rpm for 3 h. The dried powder was compressed into a pellet and sintered at 1200 °C for 15 h. LLZ ceramics were characterized by powder X-ray diffraction (XRD) measurements for structural analysis. XRD data were collected using a Stoe Stadi-P Image Plate, IP, (Stoe and Cie GmbH, Darmstadt). Data were collected over 0 ≤ 2θ ≤ 100 using monochromatic Cu Kα₁ radiation (λ = 1.5406 Å). Impedance spectroscopy (IS) measurements were performed on the sintered materials using Novocontrol concept 50 system in the 1–10⁷ Hz frequency range. The IS measurements were performed in the 160–400 K temperature range where the temperature was controlled by the Quatro cryosystem in dry nitrogen atmosphere.

3. Results and discussion

The XRD pattern of the sintered LLZ sample is shown in Fig. 1. XRD pattern corresponds to the cubic LLZ with no extra peaks are observed. The electrical properties of the investigated materials have been studied through impedance spectroscopy measurements. Complex impedance diagrams of LLZ ceramics at selected temperatures are shown in Fig. 2. The impedance data show one semicircle at high frequencies followed by a large spike at low frequencies due to electrode polarization effect. It is clear from this figure that the impedance semicircle cannot be resolved to grain and grain boundary contributions; therefore the intercept of the semicircle with the real axis represents the total ionic conductivity. The temperature dependence of the Li⁺ conductivity of the LLZ ceramics is shown in Fig. 3, which is described by the Arrhenius relation:where σ_o is the pre-exponential factor, k is Boltzmann's constant and ΔE is the activation energy for the ionic conduction. LLZ has a total ionic conductivity value of 3.01 × 10⁻⁴ S cm⁻¹ at 25 °C with an activation energy value of 0.33 eV. This value of the ionic conductivity is the same as reported by Murugan et al. despite they have sintered the sample at a higher temperature of 1230 °C for a longer duration of 36 h.⁴

Fig. 1 Powder X-ray diffraction pattern of Li₇La₃Zr₂O₁₂ together with the standard pattern of Li₅La₃Ta₂O₁₂ (PDF # 80-0458).

Fig. 2 Complex impedance diagrams of Li₇La₃Zr₂O₁₂ at selected temperatures.

Fig. 3 The temperature dependence of the dc conductivity of Li₇La₃Zr₂O₁₂ garnet material.

Regardless of the vast number of studies on structure and transport properties of LLZ ceramics, there are no estimates of the concentration of mobile Li⁺.^4,12–21 Thangadurai and Murugan and co-workers have routinely calculated the total density of Li⁺, N, in garnet materials and used it as the true value of the concentration of mobile Li⁺, n_c, that participate in the conduction process.^9,10,21 Due to the complex distribution of Li⁺ between tetrahedral and octahedral sites, it is expected that a fraction of Li⁺ could be immobile,^22,23 then the value of n_c is expected to be smaller than that of N. In this work we estimate the values of n_c, the corresponding hopping frequency and the mobility of mobile Li⁺ from the analysis of the conductivity spectra at different temperatures.^30–32

The frequency dependence of the real part of the conductivity, σ′(ω), for LLZ ceramics is shown in Fig. 4 at selected temperatures. At low temperatures and frequencies, random diffusion of the ionic charge carriers via activated hopping gives rise to a frequency-independent dc conductivity, σ_dc. With increasing frequency, σ′(ω) shows a dispersion that shifts to higher frequencies with increasing temperature. An additional feature is observed at high temperatures where σ′(ω) decreases at lower frequencies due to space charge polarization at the blocking electrodes. The conductivity spectra are usually analyzed by a power-law model of the form,³³

σ′(ω) = σ_dc[1 + (ω/ω_c)ⁿ], (3)

where σ_dc is the dc conductivity, ω is the angular frequency, n is the power-law exponent, and ω_c is the crossover radial frequency from the dc to the dispersive conductivity region. The dc conductivity in eqn (1) could be re-written in the form;where γ is a geometrical factor for ion hopping (γ = 1/6 for isotropic conduction process), λ is the hopping distance, and ω_H is the hopping frequency of mobile ions. The crossover frequency ω_c usually represents the true hopping frequency, ω_H, of mobile ions.^30–32 Therefore, we have analyzed the conductivity spectra of the investigated materials using eqn (3) in order to determine the values of σ_dc and ω_c.

Fig. 4 Conductivity spectra at selected temperatures for Li₇La₃Zr₂O₁₂ lithium conducting garnets. The solid curves between the points are the fits to eqn (3).

Both the concentration n_c and the hopping frequency ω_H of mobile Li⁺ may be thermally activated and could be written as;

where E_c and E_H are the activation energy for the creation and migration of charge carriers, respectively. It is observed from eqn (4) and (5) that the activation energy of the dc conductivity is E_σ = E_c + E_H. The fitting results of the conductivity data according to eqn (3) are shown as solid curves in Fig. 4, and the extracted values of σ_dc, ω_c and n are summarized in Table 1. The fitting process is limited to low temperatures <230 K because the conductivity dispersion region shifts out of our frequency window at higher temperatures. The reciprocal temperature dependence of the dc conductivity σ_dc and the hopping frequency ω_H obtained from the fitting process is shown in Fig. 5. The values of the activation energy for the ionic conduction, E_σ, and for ion hopping, E_H are 0.34 and 0.36 eV, respectively. The close agreement between E_σ and E_H suggests that the concentration of mobile ions, n_c, is independent of temperature (i.e. E_c ∼ 0).^30–32

Table 1 The dc conductivity σ_dc, the hopping frequency ω_H and the power-law exponent n at different temperatures as determined from the fitting of the conductivity spectra. n_c, μ and D are the concentration, the mobility and the diffusion coefficient of mobile Li⁺ as calculated from eqn (4) and (6)

T (K)	σdc (S cm^−1)	ωH (Hz)	n	nc (cm^−3)	μ (cm^2 V^−1 s^−1)	D (cm^2 s^−1)
170	1.07 × 10^−8	6.06 × 10^3	0.53	3.36 × 10^21	1.99 × 10^−11	2.92 × 10^−13
180	3.73 × 10^−8	2.38 × 10^4	0.53	3.15 × 10^21	7.40 × 10^−11	1.15 × 10^−12
190	1.17 × 10^−7	8.29 × 10^4	0.53	2.98 × 10^21	3.48 × 10^−10	3.99 × 10^−12
200	3.22 × 10^−7	2.21 × 10^5	0.51	3.25 × 10^21	6.19 × 10^−10	1.07 × 10^−11
210	8.44 × 10^−7	6.34 × 10^5	0.51	3.12 × 10^21	1.69 × 10^−9	3.05 × 10^−11
220	2.00 × 10^−6	1.52 × 10^6	0.50	3.22 × 10^21	3.87 × 10^−9	7.34 × 10^−11
230	4.47 × 10^−6	3.72 × 10^6	0.51	3.08 × 10^21	9.05 × 10^−9	1.79 × 10^−10

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Fig. 5 The temperature dependence of the dc conductivity (closed symbols) and the hopping frequency (open symbols) determined from the fitting of the conductivity spectra for Li₇La₃Zr₂O₁₂.

We have estimated the values of n_c for mobile Li⁺ in LLZ ceramics using eqn (4), where a hopping distance of λ = 1.7 Å has been used.³⁴ The estimated values of n_c at different temperatures are listed in Table 1. We notice that the values of n_c are independent of temperature with an average value of 3.17 × 10²¹ cm⁻³. The temperature independent values of n_c indicate that the mobility of Li⁺ is the factor that controls the conduction process in LLZ garnets.

It is interesting mentioning that Thangadurai and Murugan and co-workers have reported the concentration of Li⁺ in different garnet materials.^9,10,21 However, in their studies they have used the value of the total density of Li⁺, N, as the true concentration of mobile Li⁺ that are involved in the conduction process. The total density is calculated from the relation; N = m/V, where m is the number of Li⁺ per unit cell and V is the volume of the unit cell.^9,10,21 This relation gives values of N ranging from 1.90 × 10²² to 2.57 × 10²² cm⁻³ for different garnet materials. Clearly these values of N are much higher than the true concentration of mobile Li⁺ as indicated in the current work. Using the value of N in place of n_c lead to erroneous estimates of the mobility and diffusivity of Li⁺.^9,10,21 In order to estimate the attempt frequency of Li⁺ in Li₇La₃Hf₂O₁₂, Zaiβ et al. have assumed a value of n_c = 3.2 × 10²¹ cm⁻³, which agrees with our current estimates for LLZ. However, they did not give any information how they have estimated n_c.³⁵ The current work is quite different from the previous studies, where we have systematically investigated the true concentration of Li⁺ at various temperatures and estimated the true mobility/diffusivity of mobile ions.

Here we compare the values of n_c with that of the total density N of Li⁺ in LLZ. Using a value of m = 56 for LLZ garnets and a lattice parameter of 12.9699 Å,⁴ gives a value of N = 2.57 × 10²² cm⁻³. The percentage of the concentration of mobile Li⁺, n_c, out of the total Li⁺ density, N, in LLZ garnets is only 12.3%. This means that only a small fraction of Li⁺ participate in the conduction process in LLZ garnets.

Several experimental and computational studies have been reported to elucidate the Li⁺ ionic conduction pathways in garnet materials. Wullen et al. concluded from NMR data that Li⁺ in 24d sites are immobile, and the conduction takes place between octahedral sites only.²² Xu et al. using nudged elastic band method have suggested two different Li⁺ pathways; routes A and B.²⁹ In route A, with high energy barrier of 0.8 eV, Li⁺ migrate from one octahedral site to the next octahedral site bypassing the tetrahedral site, whereas in route B, with lower energy barriers of 0.26 eV, the conduction pathway involves the tetrahedral sites.²⁹ It is assumed that route B becomes more probable with increasing Li⁺ content such as in LLZ garnets.²⁹ Adams and Rao from their XRD and neutron diffraction and molecular dynamics simulations studies showed that the lower energy pathways for Li⁺ conduction involve both 24d and 96h sites in LLZ garnets, where four local 24d–96h–96h–24d paths are interconnected at the 24d site to form a 3D network of pathways.³⁴ Using high temperature neutron diffraction coupled with maximum entropy method Han et al. confirmed a 3D diffusion pathway of Li⁺ consisting of interlocking 24d–96h–48g–96h–24d chain segments.²⁷ More recently, Wang et al. through NMR study of Al and Te co-doped LLZ confirmed the 24d–96h–48g–96h–24d diffusion pathway, and assumed that the mobility of Li⁺ in 24d sites is the determining factor for the ionic conductivity.³⁶

According to the occupancy of the octahedral and tetrahedral sites in LLZ [∼90% and 56%, respectively (ref. 28 and 29)], the concentration of the octahedral and tetrahedral vacant sites is 2.2 × 10²¹ and 4.84 × 10²¹ cm⁻³, respectively. The concentration of the vacant octahedral sites is less than the estimated concentration of mobile Li⁺ of 3.17 × 10²¹ cm⁻³. This means that the vacant octahedral sites cannot accommodate all of the mobile Li⁺ in LLZ, which confirms that the transport process involves the diffusion of Li⁺ in both octahedral and tetrahedral sites in the cubic LLZ garnets.

The diffusion coefficient of Li⁺ is related to the mobility and the dc conductivity through the relation;

The values of the mobility μ, calculated from eqn (1), and the diffusion coefficient D of Li⁺ are listed in Table 1 at different temperatures. Extrapolation of the diffusion coefficient of LLZ to RT (303 K) gives a value of D of 1.33 × 10⁻⁸ cm² s⁻¹, which is comparable with other fast Li⁺ conductors.³⁷ It is interesting to compare the various parameters of the ionic transport of garnet phases with different Li⁺ content. Recently we have studied the transport and electrical relaxation properties of LLT and LBLT garnets.³⁸ We show in Fig. 6 the temperature dependence of the ionic conductivity and the diffusion coefficient for LLT, LBLT and LLZ garnets. The concentration of mobile Li⁺ in LLT and LBLT is 6.71 × 10²⁰ and 1.92 × 10²¹ cm⁻³, respectively, which represents 3.5% and 8.8% out of the total Li⁺ density in these materials.³⁸ At 220 K, for example, the ionic conductivity value is 1.33 × 10⁻⁹, 1.27 × 10⁻⁷ and 2.0 × 10⁻⁶ S cm⁻¹ and the diffusion coefficient value is 2.65 × 10⁻¹³, 7.59 × 10⁻¹² and 7.34 × 10⁻¹¹ cm² s⁻¹ for LLT, LBLT and LLZ, respectively.³⁸ We notice from these results that both the concentration and diffusivity/mobility of mobile Li⁺ increase with increasing the Li⁺ content per unit formula in garnet materials, leading to the enhanced ionic conductivity. However, the impact of the mobility is much more prominent than n_c in lithium conducting garnets.

Fig. 6 The temperature dependence of (a) the dc conductivity and (b) the diffusion coefficient for Li₇La₃Zr₂O₁₂, Li₆BaLa₂Ta₂O₁₂ and Li₅La₃Ta₂O₁₂ garnet materials.

From the current and previous results we can summarize the following points: (i) the total density N of Li⁺ in garnet materials is already high in the range of 1.9 × 10²² to 2.57 × 10²² cm⁻³ depending on Li⁺ content, (ii) only small fraction of 3.5–12.3% of Li⁺ are mobile in garnet materials with different Li⁺ content, (iii) in low Li⁺ content garnets such as LLT, with 5 Li⁺ per unit formula, Li⁺ in 24d sites are immobile and are not involved in the conduction process,²² which hinder the mobility of Li⁺ in 48g/96h octahedral sites leading to low Li⁺ mobility and low conductivity, (iv) increasing Li⁺ content lead to re-arrangement of Li⁺ ions, creates more vacant 24d sites and stimulate the mobility of more Li⁺ in these 24d sites which facilitate the 3D 24d–96h–48g–96h–24d diffusion process, such as in LLZ garnets,³⁶ (v) with increasing the Li⁺ content from LLT to LLZ (with a three orders of magnitude enhancement of the conductivity at low temperatures) the value of n_c increases only by a factor of 4.7, whereas the diffusivity increases considerably by a factor of ∼276. This means that increasing Li⁺ content per unit formula from 5 to 7 in LLT and LLZ garnets, respectively lead to minimal increase of n_c but considerable enhancement of the mobility/diffusivity of mobile Li⁺. From this summery we suggest that for further enhancing the ionic conductivity in garnet materials we need to fine tune the occupancy of tetrahedral and octahedral sites so that more vacant 24d and 48g/96h sites are created and to stimulate the mobility of Li⁺ in 24d sites.

4. Conclusions

Li₇La₃Zr₂O₁₂ lithium conducting garnets have been prepared by conventional solid state reaction route. LLZ has ionic conductivity value of ∼3 × 10⁻⁴ S cm⁻¹ at RT with activation energy of 0.33 eV, in agreement with earlier studies. We have studied the ionic transport properties through the analysis of the conductivity spectra at different temperatures in order to extract the factors that control the ionic conduction process. The extracted dc conductivity and hopping frequency are thermally activated with the same activation energy. We have estimated the true concentration of mobile Li⁺ at different temperatures and found that n_c is independent of temperature with an average value of 3.17 × 10²¹ cm⁻³. This value represents only 12.3% of the total Li⁺ content in LLZ garnet material. The value of n_c is larger than the density of vacant octahedral sites, which indicates that the conduction process involves the diffusion of Li⁺ in both the octahedral and tetrahedral sites. The enhanced ionic conductivity in LLZ compared to LBLT and LLT garnets with lower Li⁺ content is due to the increased concentration and mobility of mobile Li⁺, with the mobility having much prominent influence than n_c. The mobility and diffusivity of mobile Li⁺ are estimated at different temperatures, with estimated values comparable to other fast Li⁺ conductors.

Acknowledgements

The author acknowledges the financial support from King Abdulaziz City of Science and Technology (KACST) under grant no. ARP-30-109.

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