Lithium-7 NMR and ionic conductivity studies of lanthanum lithium titanate electrolytes

Solid State Ionics 99 (1997) 41–51

Lithium-7 NMR and ionic conductivity studies of lanthanum lithium titanate electrolytes a a b, b J. Emery , J.Y. Buzare , O. Bohnke *, J.L. Fourquet a

Laboratoire de Physique de l’ Etat Condense´ ( UPRESA 6087 CNRS), Universite´ du Maine, Avenue O. Messiaen, 72085 Le Mans Cedex 9, France b Laboratoire des Fluorures ( UPRESA 6010 CNRS), Universite´ du Maine, Avenue O. Messiaen, 72085 Le Mans Cedex 9, France Received 21 November 1996; accepted 12 March 1997

Abstract Lanthanum lithium titanate compounds belonging to the solid solution (La 2 / 32x Li 3x h 1 / 322x )TiO 3 have been investigated by powder X-ray diffraction analysis, electrical conductivity and 7 Li nuclear magnetic resonance (NMR) spectroscopy. These materials are purely ionic conductors with a maximum ionic conductivity of 3.9 3 10 24 S cm 21 at room temperature found for x 5 0.08. 7 Li NMR and room temperature conductivity measurements indicate that there are two kinds of lithium ions with different environments and different mobilities present in these materials. This result is in agreement with the crystal structure established by powder X-ray diffraction analysis. Different values of the activation energy have been obtained from spin–lattice relaxation time T 1 measurements at low temperatures and from conductivity measurements (0.12 eV and 0.37 eV respectively). Moreover, the characteristic vibration frequency for the ionic hopping motion obtained from T 1 measurements (n0 ¯ 10 11 Hz) is an order of magnitude lower than that of typical phonon frequency, namely, 10 12 –10 13 Hz. These results may suggest that the motion of the mobile lithium ions is highly correlated and / or that the dimensionality of the motion has a 2D character. Keywords: 7 Li NMR; Ion conductivity; Electrolyte – titanate

1. Introduction Studies of solid electrolytes with high lithium ionic conductivity, i.e. 10 23 S cm 21 at 300 K, have been reported recently by Inaguma et al. [1–4]. These compounds have the general formulation (La 2 / 32x Li 3x h 1 / 322x )TiO 3 where h is an A-site vacancy. They present a perovskite-type structure ABO 3 . The presence of cation deficiencies at the *Corresponding author. Tel: 133-2-4383-3354; fax: 133-24383-3506; e-mail: [email protected] 0167-2738 / 97 / $17.00  1997 Elsevier Science B.V. All rights reserved PII S0167-2738( 97 )00202-6

A-sites are favourable for high mobility of lithium ions through the bottleneck formed by four adjacent BO 6 octahedra. A careful study of the La 2 O 3 -Li 2 O-TiO 2 ternary system has been performed recently by Robertson et al. [5]. It reveals the presence of a pure phase solid solution only on the line corresponding to the series (La 2 / 32x Li 3x h 1 / 322x )TiO 3 and only in the domain 0.04 , x , 0.14 (it has to be noted that these authors use the formulation (La 0.51y Li 0.523y )TiO 3 with 0.025 , y , 0.13). More recently, Fourquet et al. carried out a structural study of these compounds by

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X-ray diffraction [6]. These authors found that the solid solution covers the same composition range as mentioned by Robertson et al. Both authors have shown that a pure solid solution does not appear to extend up to x 5 0.17 corresponding to the compound La 0.5 Li 0.5 TiO 3 as suggested more recently by the study performed by Leon et al. [7]. Fourquet et al. determined the space group, the variation of the cell parameters and the La distribution in the A-sites as a function of x. They clearly show that the structure of these compounds is characterized by some disorder in the La distribution. In order to characterize the electrical properties of these materials, conductivity measurements have been carried out in a previous study from 300 K to 700 K by using complex impedance technique [8]. In this work, variable temperature pulsed 7 Li NMR techniques have been used to further study the ionic conduction mechanism involved in these compounds. The dependence of the 7 Li NMR spectra and the spin–lattice relaxation times T 1 on both sample temperature and composition x have been determined. These data have been compared with those deduced from conductivity measurements. It has to be noted that NMR spin–lattice relaxation is a measure of the local charge-density fluctuations due to the ionic motion and that conductivity is a measure of the long-range diffusion of the charges under the effect of an electric field. Therefore the experimental data obtained by these two different techniques must be analyzed with much attention in order to obtain correct interpretations.

2. Experimental The compounds belonging to the solid solution (La 2 / 32x Li 3x h 1 / 322x )TiO 3 , that we shall name LLTO in the remainder of this paper, were synthesized from stoichiometric amounts of Li 2 CO 3 , TiO 2 and La 2 O 3 . These mixtures were heated in air at 8008C for 4 h and then fired at 11508C for 12 h. The details of the procedure have been given elsewhere [8]. At this point of the preparation, it has been shown, by chemical analysis of Li 1 , that no lithium was lost during the heat treatment at 11508C. The calcined powder was then ground, pressed into pellets of 5 mm diameter and 1–2 mm of thickness

under a pressure of 250 MPa and then sintered in air at 13508C for 6 h. The pellets were heated at a sweep rate of 108C min 21 from 200 to 13508C and then cooled down at the same sweep rate to room temperature. These pellets had a compactness of 85%. This procedure was necessary to minimize the grain boundaries resistance of the pellet. However, the chemical analysis of Li 1 shows that a Li 2 O loss occurs during the sintering process because of the high temperature used. Sintered pellets coming from the same preparation run have been used for both conductivity and NMR measurements to avoid any misunderstanding coming from a difference in the lithium content in the material due to the sintering process. The values of x mentioned in this paper refer to the exact composition of the studied compounds. The lithium content of the samples, before and after sintering, has been determined by flame emission analysis. A small amount of the sample (about 30–40 mg) was heated for 1–2 h with K 2 S 2 O 7 and dissolved in acidified water. The accuracy of the measurement was around 3%. Conductivity measurements were carried out by the complex impedance method from 300 K to 700 K under N 2 atmosphere using ionically blocking electrodes (Pt) in the 32 MHz–1 Hz frequency range. A 100 mV (r.m.s) applied voltage was used. Details of the procedure used are given in Ref. [8]. Static 7 Li NMR spectra were recorded with a MSL / 300 spectrometer at resonance frequency of 116 MHz by using a simple p / 2-acquisition sequence with phase cycling at room temperature. Spin–lattice relaxation times T 1 have been obtained by using an inversion recovery sequence (p-t -p / 2acquisition) in the temperature range 160 K–400 K with a 10 K step. To improve the signal to noise ratio the sequences were accumulated either 64 or 128 times. To determine the T 1 values at each temperature, recovery of experimental magnetization was fitted to the exponential function: M(t ) 5 M0 [1 2 2a exp(2t /T 1 )]

(1)

where the spin–lattice relaxation time T 1 , the thermal equilibrium magnetization M0 and a were considered to be free parameters in a least-squares fitting procedure. 7 Li magic-angle-spinning (MAS) spectra were

J. Emery et al. / Solid State Ionics 99 (1997) 41 – 51

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recorded at room temperature using a MSL / 300 spectrometer at a spinning rate of 10 kHz.

3. Results

3.1. Powder X-ray diffraction analysis We will recall briefly here the main results of a previous study performed on LLTO samples before sintering [6]. Fig. 1 presents the crystal structure of LLTO. The space group is P4 / mmm. The structure is characterized by some disorder in the La, Li and vacancies distribution among the sites 1a (0,0,0) and 1b (0,0,1 / 2). X-ray diffraction patterns were indexed ˚ and c52a. in a tetragonal cell with a5b¯3.87 A The presence of superstructure lines observed in these patterns implies the doubling of the c-axis. Moreover, it has been observed that the tetragonal distortion c / 2a begins to disappear at x$0.08. Fig. 2 a and b show the evolution of La distribution (in % occupancy of the site) over the two sites 1a and 1b respectively as a function of composition. La 31 ions are preferentially distributed over the 1a site with a maximum value of % occupancy at x50.08. Because of the low diffracting power of Li 1 ions, it was not

Fig. 2. Evolution of La distribution over the site (0,0,0) (curve a) and the site (0,0,1 / 2) (curve b) as a function of x in La 2 / 32x Li 3x TiO 3 .

possible to determine the Li and, consequently, the vacancies distributions over 1a and 1b sites.

3.2. Chemical analysis of the sintered compounds

Fig. 1. Crystal structure of La 2 / 32x Li 3x TiO 3 [s5La, Li or vacancies in position (0,0,0), + 5La, Li or vacancies in position (0,0,1 / 2)]. Lines represent the unit cell.

The chemical analysis of lithium in the sintered samples shows a lithium loss of about 20 to 30% in lithium atom % during sintering, depending on the composition of the LLTO. The higher the lithium content, the higher the lithium loss. Table 1 gives the Li content in the compounds before and after sintering as well as the atom% loss. Powder X-ray diffraction analysis carried out on

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Table 1 Chemical analysis data for LLTO before and after sintering Before sintering

After sintering

atom% loss

x

Analysis data

x (60.003)

Analysis data

0.17 0.13 0.11 0.09 0.07

not a pure solid solution Li 0.39 La 0.53 TiO 3 Li 0.33 La 0.55 TiO 3 Li 0.27 La 0.57 TiO 3 Li 0.21 La 0.59 TiO 3

0.120 0.095 0.080 0.065 0.057

Li 0.36 La 0.55 TiO 3 Li 0.29 La 0.57 TiO 3 Li 0.24 La 0.59 TiO 3 Li 0.20 La 0.60 TiO 3 Li 0.17 La 0.61 TiO 3

29 26 27 28 19

LLTO after sintering shows that no structural change is observed but only a small variation of the unit cell parameters: the loss of Li 2 O during sintering leads to the formation of a new compound of the solid solution with less Li content. Therefore some impurities might be present in the solid solution due to the loss of Li 2 O and consequently to the formation of TiO 2 and La 2 O 3 . However these impurities are present in very small quantities (about 3% in maximum) and can not be detectable by X-ray diffraction analysis.

3.3. Ionic conductivity measurements It has been shown in a previous study that LLTO compounds can be considered as purely ionic conducting materials [8]. Indeed, the maximum total conductivity, i.e. ionic and electronic, at T5300 K has been found, for the bulk material, to be st 5 3.9310 24 S cm 21 for x50.08 and the electronic conductivity se 55310 210 S cm 21 at the same temperature. Fig. 3 presents the dc-conductivity data, expressed as Ln(s T ) versus inverse temperature for x50.057 and x50.08. It can be observed that the conductivity data do not follow an Arrhenius law over all the investigated temperature range. For temperatures lower than 400 K, conductivity data follow an Arrhenius law with an activation energy of 0.37 eV. The mechanism of conduction is thermally activated. On the other hand, for temperatures higher than 400 K, conductivity data can be fitted to a Vogel–Tamman–Fulcher (VTF)-type relationship. The mechanism of conduction becomes thermally assisted. It may involve the tilting of the BO 6 octahedra due to the increase of temperature which influences the motion of the ions since the mobile ions have to move through the bottleneck formed by

Fig. 3. Logarithmic plot of the total conductivity as a function of inverse temperature T for La 2 / 32x Li 3x TiO 3 (d) x50.057 and (n) x50.08.

four adjacent octahedra. The pseudo activation energy decreases to 0.06 eV. The variation of room temperature dc-conductivity as a function of the solid solution composition x is shown in Fig. 4. As previously observed by Inaguma et al. [2] and by Kawai et al. [9], the conductivity increases, reaches a maximum value and decreases as the lithium content increases. The ionic dc-conductivity s is given by the general formula:

s 5 qnm

(2)

where q is the charge, n the number and m the mobility of the mobile species in the conductor. In

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45

ions present in this site may become more mobile and the vacancies more available. This phenomenon may explain that the conductivity does not follow Eq. (3) for small x values. From the conductivity, the chemical diffusion ˜ can be obtained from the Nernst– coefficient, D, Einstein relationship:

Fig. 4. Room temperature dc-conductivity (s), the average splitting of the satellite lines of Fig. 5 kCq l (d) and the constant C obtained from the BPP Eq. (5) (h) as a function of x in La 2 / 3-x Li 3x TiO 3 .

LLTO compounds the mobile species are Li 1 ions. Since they are expected to move through the A-site vacancies, their mobility might be proportional to the fraction of these A-site vacancies. The number of vacancies is expressed as (1 / 322x) and the number of lithium ions as 3x. Therefore the ionic conductivity might be proportional to:

s ~(3x)(1 / 3 2 2x).

s RT D˜ 5 ]] . (4) nF 2 If we assume that all the lithium ions contribute to the conduction (for x.0.08, as found previously), then n53.4310 23 mole cm 23 for x50.08. The 28 ˜ chemical diffusion coefficient is then D53310 2 21 cm s at 300 K, which has also been determined by impedance spectroscopy during lithium intercalation. A value of 5310 28 cm 2 s 21 has been obtained before intercalation. The two obtained values are in good agreement but, in the later case, the uncertainty is high owing to the difficulty in determining the slope of the curve dE / dx at the beginning of the intercalation with good accuracy [8]. 3.4. NMR study Fig. 5 presents typical static NMR spectra of 7 Li obtained at room temperature by using a phase

(3)

This equation explains qualitatively the experimental curve of Fig. 4 only for x.0.08. For smaller values of x it seems that all the lithium ions are not mobile in the structure and / or that all the A-sites vacancies are not available for ionic conduction path. This result can be compared to the La occupancy of the two sites (0,0,0) and (0,0,1 / 2). For small values of x (x,0.09, see Fig. 2) the La occupancy of site (0,0,0) is very high and therefore the Li 1 ions which occupy this site may be not mobile and / or the A-site vacancies in this site may not be available for long range conduction. On the other hand as soon as this site is less occupied by La 31 ions (x.0.09), the Li 1

Fig. 5. Composition dependence of the static 7 Li NMR spectra recorded at 116 MHz and at room temperature.

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cycling single pulse sequence (p / 2-acquisition) on four LLTO samples. No selective excitation was used and then the spectra account for the quadrupolar character of the 7 Li nuclei with spin 53 / 2. In this case and for all the studied compositions, the observed spectra exhibit a narrow and intense line, coming from the 21 / 2↔1 / 2 transition, and two broad satellite lines coming from the 1 / 2↔3 / 2 and 21 / 2↔23 / 2 transitions associated with the electric quadrupolar interactions. These satellites have not been observed by Leon et al. in the compound La 0.5 Li 0.5 TiO 3 (7). A main feature revealed by these spectra is the non agreement of the relative intensities between the central line and the satellites suggesting that the observed spectra are more complicated than the static NMR spectrum of 7 Li usually formed by a central line and by two satellite lines. Therefore three individual mechanisms can be put forward to explain the particular shape of these spectra: (i) two kinds of lithium ions with different mobilities exist in the structure, each of them giving rise to different spectra corresponding to different quadrupolar splitting and leading to the enhancement of the central line; (ii) the presence of some disorder effects in the structure: in this case a static distribution of the quadrupolar splitting has to be taken into account, this distribution acts at second order on the central line (leading to the enhancement of this peak) and also broadens the satellites, so the central line would be enhanced again; (iii) only one kind of mobile lithium ions in an intermediate dynamical regime exists in the material, in such a case the narrow central line would undergo a narrowing effect while the satellites would undergo a broadening effect. We have to point out that one or several of these mechanisms may account for the particular shape of the spectra. However at this point it is particularly difficult to differentiate the different contributions. We can observe in Fig. 5 that the central line is relatively narrow (of the order of 400 Hz at room temperature for x50.08, for example). Such a width is of the same order of magnitude as the value obtained with liquid or gel lithium electrolytes whose conductivity is of the order of 10 22 to 10 23 S cm 21 at room temperature [10]. This result suggests that the Li 1 ions have a very high mobility in these compounds. It is in agreement with conductivity data obtained previously [8].

We can also observe in Fig. 5 that the satellites splitting varies with the x values: this splitting decreases from x50.057 to x50.08 and increases between x50.08 and x50.095. Such a behaviour can be compared with the variation of the dc-conductivity with x, as shown in Fig. 4. The average splittings between the satellites have been obtained by roughly fitting the experimental spectra of Fig. 5. Because of the width of the satellite lines, the quadrupolar spectra were fitted with a very large peak. Therefore the obtained values, which will be noted kC q l, have to be considered as approximated values. So, we can reasonably think that the non agreement of the relative intensities between the central line and the satellites is due to the superposition of dynamical effects that leads to a distribution in the intermediate regime and maybe is also due to some disorder effect. The dynamical aspect is supported by temperature studies which show that the satellites are better resolved at high temperatures as shown in Fig. 6 for x50.08. Fig. 7 presents a static NMR spectrum (curve a) and a Magic Angle Spinning (MAS) spectrum (curve b) of 7 Li obtained at room temperature for x50.057. As previously mentioned the non agreement of the relative intensities between the central line and the satellites can be observed. The satellite transitions are difficultly observed in the static spectrum but

Fig. 6. Temperature dependence of the static 7 Li NMR spectra recorded at 116 MHz for x50.08.

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Fig. 7. (a) Static 7 Li NMR spectrum recorded at 116 MHz and at room temperature (dashed arrows show the smoothed singularities of the satellite transitions). (b) 7 Li MAS spectrum recorded at a spinning rate of 10 kHz and at room temperature (full arrows show the spinning sidebands related to the satellite transitions). (x50.057.)

they are evidenced in the MAS experiment through their spinning sidebands.

Fig. 8. Experimental temperature dependence of the spin–lattice relaxation time T 1 of 7 Li for the four LLTO compounds.

3.5. Spin–lattice relaxation Spin–lattice relaxation experiments have put forward the existence of two T 1 values whose behaviours have been revealed to be different with both temperature and composition. In the remainder of this paper, we will be mostly concerned with the variation of the T 1 -values associated to the central line. The evolution of the spin–lattice relaxation time T 1 as a function of the sample temperature is shown in Fig. 8 for the four LLTO compounds: x50.057, 0.065, 0.08 and 0.095. For the determination of T 1 , we took into account the more intense and narrow peak of the 7 Li NMR spectra. The small dispersion of the T 1 values observed in Fig. 8 (particularly for x50.057) may be due to some contribution of the satellite peaks to the central line and / or to the presence of the two lithium ions with two different mobilities. It is shown that below 200 K, T 1 increases drastically suggesting that the lithium ions become less and less mobile. Fig. 9 presents the T 1 -values plotted in Arrhenius fashion. This representation of the relaxation time shows clearly the temperature at which T 1 is minimum. This minimum indicates the frequency of the ionic motion. The motional correlation time, tc , is comparable to the

Fig. 9. Logarithmic plot of the spin–lattice relaxation time T 1 as a function of the reciprocal temperature for the four LLTO compounds.

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reciprocal of the angular Larmor frequency, v0 , i.e. v0tc 50.62. The presence of a well-defined minimum, observed around 350 K for the four compositions, implies that the local environment (nearest-neighbour) of the mobile ions is quite homogeneous in all the compounds. It can also be observed that the relaxation time is thermally activated at low temperatures. The activation energy obtained from the low temperature branch of the curve Ln(T 1 ) vs. 1000 /T is approximately 0.12 eV and is different from the activation energy deduced from conductivity data (0.37 eV). In a first approximation, by using the Bloembergen–Purcell–Pound (BPP) [11] model which assumes an isotropic motion of particles and an exponential correlation function, the spin–lattice relaxation time, T 1 , can be related to the correlation time of the ionic motion, tc , by the relationship: 1 ]5C T1

F

tc 4tc ]]] 2 1 ]]]] 1 1 (v0tc ) 1 1 (2v0tc )2

G

(5)

where C is a constant which accounts for the microscopic mechanism of the relaxation process. If this process is due to pure quadrupolar effects, C is related to the quadrupolar parameter Cq by the relationship:

S

1 h2 C 5 ](Cq )2 1 1 ] 10 3

D

(6)

and 3Cq Cq e 2 QVzz ]]] ] ]] nq 5 5 , Cq 5 , 2 " 2I(2I 2 1)

(7)

where Q is the quadrupolar moment of 7 Li, Vzz is the nucleus electric field gradient, h the asymmetry parameter and " the Planck’s constant. Since the BPP model represented by Eq. (5) predicts a minimum of T 1 for v0tc 50.62 [11], the constant C can be evaluated from both the experimental data of T 1 at the curve minimum and Eq. (5). A C-value which varies from 6310 10 s 22 to 13310 10 s 22 , depending on the composition of the material, has been determined. These values have been reported in Fig. 4. The C-value is ascribed to the quadrupolar relaxation due to charge fluctuations caused by lithium motion. They agree with a quadrupolar frequency nq of the order of 500 kHz, according to Eq. (7). However

these values can not be compared to the average splitting of the satellite lines, kCq l obtained by a rough fitting of the satellites position, shown in Fig. 4, which varies from 40 to 20 kHz depending on the composition x. Moreover we can observe in Fig. 4 that the experimental values of both the C and the s -dc present a maximum for x50.08, although the kCq l-values present a minimum. These discrepancies observed between both the values and the variations of C (or Cq ) and kCq l merit some comments: the Cq -values are obtained from a dynamical equation (Eq. (5)) while the kCq l-values are obtained by using a static Hamiltonian and corresponds to an average value which does not take into account the motion effects. These remarks suggest that Cq and kCq l are not related to the same mechanism and / or to the same particles (two types of lithium ions). We can postulate that a dynamic effect has to be taken into account in the satellite lines, that another relaxation mechanism can be involved in the process of conduction and / or that two kinds of lithium ions with different mobility may be involved in the satellite lines and in the central line. It is possible from the dependence of T 1 versus temperature to determine the thermal dependence of the correlation time tc of the motion (or the frequency jump nc ) with temperature, as shown in Fig. 10 for x50.08. The frequency jump is thermally activated at low temperatures and can be fitted to an Arrhenius-type law as:

S

Ea nc 5 n0 exp 2 ] kT

D

(8)

with Ea 50.12 eV and n0 57.4310 10 Hz. A good agreement is obtained at low temperatures for the activation energy deduced from the T 1 values and the tc values. This is not surprising since for the temperatures below the temperature of the minimum of T 1 , the BPP model predicts that T 1 ~tc . Furthermore, in this temperature range, T 1 does not depend on the dimensionality of the movement. On the other hand, the departure of the curve in Fig. 8 from Arrhenius law at high temperatures (for 1000 /T ,3 K 21 ) may be ascribed to a dimensionality and / or to a correlation effect of the Li ions motion. NMR experiments have to be performed at higher temperatures to yield information on this point.

J. Emery et al. / Solid State Ionics 99 (1997) 41 – 51

Fig. 10. Temperature dependence of the 7 Li jump frequency obtained from BPP analysis of the spin–lattice relaxation time T 1 of Eq. (5). The line represents the Arrhenius-type law fitting.

4. Discussion As suggested by powder X-ray diffraction analysis, La 31 ions are distributed in both (0,0,0) and (0,0,1 / 2) sites in the structure of LLTO (Fig. 1). It can also be determined from these spectra that the La 31 distribution among the different sites depends on the x values. For example, for low values of x (x#0.09), La 31 ions are preferentially distributed in the (0,0,0) site with 90% of site occupancy (Fig. 2 a). On the other hand only 30% of the (0,0,1 / 2) site is occupied by these ions (Fig. 2 b). These results have been confirmed by neutron diffraction analysis. However we were not able to determine with accuracy the position and the distribution of both the Li 1 ions and the vacancies neither by X-ray diffraction nor by neutron diffraction experiments. Nevertheless it seems reasonable to assume that the Li 1 ions and the vacancies are distributed among or near these two A-sites with only a small fraction in (or near) the (0,0,0) site. It can then be postulated that at least two kinds of lithium ions exist in these compounds with

49

different environments. One kind of lithium is present in (or near) the (0,0,0) site and preferentially surrounded by La 31 ions, and another kind of lithium is present in (or near) the site (0,0,1 / 2) and surrounded by either La 31 ions, other Li 1 ions and / or vacancies. These different environments may lead to different mobilities of the Li 1 ions and may explain the NMR spectra which show the presence of different lithium in the structure and certainly different mobilities. It has also been pointed out by NMR that the mobile lithium has a homogeneous local environment. Conductivity data follow an Arrhenius law below 400 K as found by NMR experiments. However, the activation energy deduced from conductivity data and from NMR measurements are different, i.e. 0.37 eV and 0.12 eV respectively. This discrepancy can be due to the fact that NMR spin–lattice relaxation is a measure of the local motion of the lithium ions. Therefore, the activation energy obtained by this technique (E NMR ) refers to a microscopic energy a micros barrier, E a , that the ions have to overcome in order to move through the bottleneck of the perovskite structure. On the other hand, conductivity is a measure of the long-range diffusion of these charges under the effect of an applied electric field. Consequently, the activation energy may be related to a macroscopic activation energy and then E sa 5E macros . a Ngai has suggested that the microscopic activation energy may be related to the macroscopic activation energy by using the Kohlrauch exponent b through the following equation [12]: E microsc E aNMR a b 5 ]] . macros 5 ]] Ea E sa

(9)

The origin of this exponent is most often interpreted as a result of a correlation between the ions during diffusive motion although it is always a controversial topic. Ngai’s model has been successfully used in the case of glassy fast-ion conductors by Reau et al. [13]. The use of this model in the case of LLTO materials leads to a value of b of 0.32 suggesting that the ionic motion is then highly correlated. The measure of relaxation time at high temperatures as well as the measure of ac conductivity in the complex modulus formalism could lead to the independent determination of the Koh-

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lrauch exponent and then could confirm the correlation of the ionic movement. However ac conductivity measurements have to be performed at temperatures lower than the ambient temperature to obtain the maximum of the imaginary part of the modulus in the frequency domain used in our experiments. These experiments are currently being performed. The correlation time tc is linked to the diffusion coefficient D˜ by the relationship: l2 D˜ 5 ] ntc

(10)

where l is the length jump and n56 if we assume an isotropic diffusion in the three directions. Since the Li 1 ions are in (or near) the A-sites of the perovskite and if we assume that they are mobile only through the vacant adjacent A-sites, the length jump can be evaluated from the structural properties of the oxide ˚ At room temperature, tc does not as l5a¯3.87 A. vary much with composition. It is found to be close to 9310 210 s. The diffusion coefficient D˜ calculated from the above relationship is then found to be 2.8310 27 cm 2 s 21 . This value is slightly higher than the value of the chemical diffusion coefficient obtained from impedance spectroscopy measurements and conductivity (i.e. from 3310 28 to 5310 28 cm 2 s 21 for x50.08 and x50.11 respectively). This result may suggest that the Li 1 ion does not jump from its A-site to the nearest vacancy A-site but to another site in the cage closer to its initial position. The preexponential factor, n0 in Eq. (8), is obtained from the BPP model which assumes an isotropic motion. This factor does not vary much with composition. It corresponds to the characteristic vibration frequency of the mobile species. For x5 0.08, n0 57.4310 10 Hz. From a general point of view, the hopping rate of a classical particle in a potential well in terms of absolute rate theory [14] is given by an Arrhenius law with a preexponential factor of the order of a typical phonon frequency, namely, 10 12 210 13 Hz. However, the value obtained with these materials is consistently lower than the above expected value. This result is often observed in defect-structure superionic conductors such as, for instance, Li 2 Ti 3 O 7 where n0 54310 7 Hz [15], bNaAl 2 O 3 [16] or Li x La 1 / 3 Nb 12x Ti x O 3 studied by Latie et al. [17] where n0 ¯10 11 Hz. The origin of this

anomalous low prefactor is not entirely clear. Walstedt et al. suggested that ion–ion correlation may play an important role in the diffusion of ions in these materials [16]. However, we can observe that a value of ¯10 11 Hz has been found in 2D materials such as b-alumina [16] or Li x La 1 / 3 Nb 12x Ti x O 3 [17]. This preexponential factor value is very close to our own value suggesting that Li 1 ions may move into planes in the structure of these solid solution. This result would not be in disagreement with the structure of these compounds.

5. Conclusion Structural considerations as well as conductivity and spin–lattice relaxation time measurements seem to give evidence of the existence of two kinds of lithium ions with different environments and different mobilities in the (La 2 / 32x Li 3x h 1 / 322x )TiO 3 solid solution. It has been shown at low temperatures, (T ,400 K), that both the ionic conductivity and the NMR relaxation times are thermally activated. However, the activation energies deduced from these measurements are quite different, 0.37 eV and 0.12 eV respectively. Since conductivity is a measure of the long-range diffusion of the charges under the effect of the electric field, the activation energy deduced from this measurement may be considered as a macroscopic activation energy. On the other hand, NMR spin–lattice relaxation time is a measure of the local motion of the ions and consequently, the activation energy may be viewed as a microscopic activation energy. According to Ngai’s model, these two activation energies can be related by the Kolrauch exponent b. In our case the value of b 50.30 suggests that the ionic motion of the lithium ions is highly correlated in these compounds. High temperatures NMR experiments are currently being performed to allow confirmation of these assumptions.

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