Fluoride ion conduction in Pb1−xSnxF2 solid solution system

Solid State Ionics 154 – 155 (2002) 503 – 509 www.elsevier.com/locate/ssi

Fluoride ion conduction in Pb1
xSnxF2 solid solution system Shinzo Yoshikado a,*, Yoshiaki Ito b, J.M. Re´au c a

Department of Electronics, Doshisha University, Kyotanabe 610-0321, Japan Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan c Institut de Chimie de la Matiere Condensee de Bordeaux, Chateau Brivazac, 33600 Pessac, France b

Accepted 10 March 2002

Abstract The purpose of this study is to fabricate single crystals of the solid solution system Pb1
xSnxF2 (x=0, 0.1, 0.2) and to further clarify the correlation between the crystal structure and the ionic conduction by crystal structure analysis and the examination of the frequency dependence of the complex ionic conductivity using an impedance spectroscopy method. The frequency dependence was precisely measured in wide frequency and temperature ranges. The hopping frequency of fluoride ions could then be estimated. It became clear that the activation energy for mobile fluoride ion generation was very small in the system of Pb0.9Sn0.1F2 and Pb0.8Sn0.2F2, and all fluoride ions were mobile. It is suggested that the activation energy, with which fluoride ions hop to adjacent sites, is determined by the bottleneck formed by (Pb2+, Sn2+) ions. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Anion conductors; Pb – Sn solid solution; Fluoride ion conduction; Crystal structure; Impedance spectroscopy; Gate model

1. Introduction The ionic conduction mechanism in a superionic conductor is dependent on the crystal structure and the disorder of the structure induced by thermal lattice defects, nonstoichiometry and added impurities. Therefore, the distribution condition of lattice defects and local structure are fundamental issues in clarifying the ionic conduction mechanism. It has been found by many workers that the ionic conductivity of h-PbF2, of which the structure is the fluorite-type, can be increased by adding monovalent ions in the extrinsic region [1 –7]. However, it has also been reported that *

Corresponding author. Tel.: +81-774-65-6328; fax: +81-77465-6801. E-mail address: [email protected] (S. Yoshikado).

both the ionic conductivity and the lattice parameters increased with increasing amount of divalent dopant SnF2, although the concentration of fluoride ions was unchanged and the ionic radius of Sn was smaller than that of Pb [8 –16]. Lucat et al. [8], Vilminot et al. [9], Lagassie et al. [10] and Ito et al. [11] studied the ionic conductivity of Pb1
xSnxF2 (x<0.5). Kanno et al. [12], Re´au et al. [13], Perez et al. [14], Vilminot et al. [15] and Ito et al. [16] have investigated the ionic conduction and the phase relations by X-ray diffraction in PbSnF4. In the solid solution system Pb1-xSnxF2, the distribution of fluoride ions at the interstitial sites contributes to the enhancement of the fluoride ion conductivity. The purpose of this study is to gain insight on the relationship between the ionic conduction and the crystal structure through the measurements of both

0167-2738/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 ( 0 2 ) 0 0 4 8 9 - 7

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the frequency dependence of the complex ion conductivity and the X-ray diffraction of Pb1
xSnxF2 (x=0, 0.1, 0.2). Single crystals must be used to obtain the precise data of the bulk complex conductivity because the electrical conduction due to the grain boundary contributed to the measured total complex conductivity for the polycrystalline samples. Large single crystals for the measurements of the complex conductivity can be obtained for xV0.2 by the melting method. Both the frequency and temperature dependences of the dynamic conductivity in wide frequency and temperature ranges were precisely measured using impedance spectroscopy method. The activation energies for the hopping process and the generation of mobile fluoride ions were estimated from the total activation energy.

2. Experimental a-PbF2 (99.99%, Rare Metallic) and SnF2 (99.9%, Rare Metallic) were used in this study without further purification. A mixture of a-PbF2 and SnF2 was sealed in a Pt capsule in nitrogen gas for each sample. The sealed samples in the capsules were heated at about 1153 K above the melting point of PbF2 for 3 h in an electric furnace. The samples were cooled to 658 K at a cooling rate of 7.5 K/h and then kept at 658 K for 72 h. Finally, the samples were cooled to room temperature by quenching. Transparent single crystals of the PbF2 –SnF2 solid solutions (10 and 20 mol% SnF2) were synthesized and labeled Pb0.9Sn0.1F2 and Pb0.8Sn0.2F2, respectively. The grown crystals were analyzed using the single – crystal four-circle X-ray diffraction meter (Rigaku AFC5R) by measuring the integral intensity of the diffraction spot of the crystal. We assumed that the concentrations of Sn were similar to those of the mixtures prepared before the growth because the lattice parameters of several crystals sampled from the same lot were identical. The synthesized single crystal was processed into the appropriate shape with two parallel end faces to measure the ac complex conductivity. Vacuum-evaporated gold blocking electrodes were applied to the single-crystal end faces. Sample thickness was 0.7 –1 mm. A sample was loaded between the end of the inner conductor and the short terminator. Hioki 3522 (0.1 – 100 Hz), HP 4284 A (20 Hz – 1 MHz) and HP

4291 A (1 MHz –1.8 GHz) were used for measurements. The magnitude of the ac signal was adjusted such that the electrical response of the sample was linear (50 –100 mV below 1 MHz and 200 mV above 1 MHz). All measurements were carried out in the temperature range between 146 and 593 K in dry nitrogen atmosphere.

3. Results and discussion Because the angles of the main Bragg reflections for both Pb0.9Sn0.1F2 and Pb0.8Sn0.2F2 correspond to those for h-PbF2, it is concluded that the crystal structure of the former is isostructural with that of the latter. The electrical conductivity is independent on the crystal orientation because the space group is Fm3m [11]. The thermal hysteresis of the electrical conductivity has been observed only in h-PbF2. The electrical conductivity increased with increasing repetitions of the heating and cooling process and finally became constant. The frequency dependence of the real part of the measured total complex electrical conductivity rtot for h-PbF2 and Pb0.8Sn0.2F2 is shown in Fig. 1a and b, respectively. rtot is described as rtot ¼

t 1 ; S Ze þ Z b

ð1Þ

where t is the sample thickness, S is the electrode area, Ze is the electrode impedance and Zb is the bulk impedance. Ze is caused by the capacitance of the electric double layer between the electrode and the crystal. The bulk complex conductivity is described as rb ¼

t 1 S Zb

ð2Þ

Solid lines in the same figure indicate the values of rtot calculated by a least squares method using the equivalent circuit for rb shown in Fig. 2, which corresponds to the fluoride ion conduction model. It is speculated that the second plateau of the real part of rtot of h-PbF2 observed in the high-frequency region at low temperature is caused by the local motion of fluoride ions. The real parts of rtot for both Pb0.9Sn0.1F2 and Pb0.8Sn0.2F2 are dependent on the frequency in the high-frequency region. It is clear that the frequency dependence obeys the power law

S. Yoshikado et al. / Solid State Ionics 154 – 155 (2002) 503–509

505

where N is the concentration of fluoride ions, e is the elementary charge, a is the hopping distance, kB is the Boltzmann constant, c (f1) is the correlation factor, c is the site occupancy of fluoride ions and xp is the hopping angular frequency and described as
 Ea xp ¼ xp0 exp
; ð4Þ kB T where Ea is the activation energy for ionic conduction process. The temperature dependence of rfiT is described as

 E Ea þ Ec rfi T ¼ r0 exp
¼ r0 exp
; ð5Þ kB T kB T where Ec is that for the mobile ion generation process. Assuming that the same number of fluoride ions contributes to the dc and ac conductivity, the real part of rb is described as follows [20,23], 
n  x Reðrb Þ ¼ rfi 1 þ : ð6Þ xp

Fig. 1. Frequency dependence of the real part of total complex electrical conductivity of (a) h-PbF2 and (b) Pb0.8Sn0.2F2.

[C(ix)n, x is the angular frequency] at low temperatures. Such a frequency dependence is caused by the disorder of the environment around mobile ions [17] and has been observed in many materials such as alkali-priderite [18,19], Na-h-alumina [20,21] and hAgI [22]. Frequency-independent fluoride ion conductivity rfi was observed in the low-frequency region for all samples. In particular, the frequency dependence, which obeys the power law, was not obtained for h-PbF2, and such a region of was wide compared with that of other samples as shown in Fig. 1a. It was conjectured that the degree of disorder of the environment around the fluoride ion was small in h-PbF2. The frequency-independent ionic conductivity rfi is described as [20] rfi ¼

Ne2 a2 ccð1
cÞxp ; kB T

ð3Þ

xp could be determined from the frequency dependence of Re(rb) by the least squares parameter fitting method. The value of the index n was approximately 0.65 for both Pb0.9Sn0.1F2 and Pb0.8Sn0.2F2, as shown in Fig. 3, and was almost temperature independent.

Fig. 2. Equivalent circuit for the bulk complex conductivity rb of the Pb1
xSnxF2 solid solution system. ep: relative dielectric constant caused by the polarization of fluoride ions. rhf: the conductivity caused by the local motion of fluoride ions. efw: framework relative dielectric constant of a crystal.

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S. Yoshikado et al. / Solid State Ionics 154 – 155 (2002) 503–509 Table 1 Parameters for Pb1
xSnxF2 solid solution system

Fig. 3. Temperature dependence of index n of Pb0.9Sn0.1F2 and Pb0.8Sn0.2F2.

For h-PbF2, such a conduction process was not observed. Therefore, it is conjectured that the power law is caused by the disorder of the configuration of Pb and Sn atoms. The temperature dependence of rfiT and xp for Pb0.8Sn0.2F2 is shown in Fig. 4. For all samples, the bend in the Arrhenius plot was observed [24]. A similar bend was observed in Arrhenius plots of xp for both Pb0.9Sn0.1F2 and Pb0.8Sn0.2F2, and the temperature corresponding to the bend was the same. It is conjectured that above this temperature (Tb), the

Fig. 4. Temperature dependence of rfiT of Pb0.8Sn0.2F2.

Sample

E (eV)

r0 (SK/cm)

Ea (eV)

xp0 (rad/s)

h-PbF2 Pb0.9Sn0.1F2 Pb0.8Sn0.2F2

0.442 0.304 0.264

1.4104 7.6103 5.3103

0.304 0.257

1.221013 1.371013

activation energy does not increase but the number of interstitial sites, which fluoride ions can occupy, increases with increasing the temperature. Detail study is now in progress. The values of activation energies Ea and xp for the ionic conduction process below Tb are shown in Table 1. The value of Ea is almost equal to the value of the total activation energy E, and all fluoride ions are mobile because Ec=0. 3.1. Concept of activation energy described using the gate model In this section, the origin of the activation energy Ea is discussed using the hard-sphere model of activation energy based on ion size, the gate model [25]. The gate model is proposed to examine the idea that a mobile ion hops between adjacent sites only when immobile ions [gate ions, (Pb2+, Sn2+) ions] move away to give a sufficient space [25]. Assuming that hard-sphere ions vibrate around their stable sites, transition probability W is statistically – mechanically evaluated. In the Pb1
xSnxF2 system, three immobile ions construct a bottleneck (a gate). It is speculated that a mobile fluoride ion passes through a gate only when the gate is dilated because a fluoride ion has electron density at a site near the gate, as shown in Fig. 5. According to Ishii [25], mobile ion is considered to hop through the

Fig. 5. Electron density of Pb0.8Sn0.2F2 at 293 K and the configuration of fluoride ion and (Pb2+, Sn2+) ions.

S. Yoshikado et al. / Solid State Ionics 154 – 155 (2002) 503–509

path having a gate made up of three (Pb2+, Sn2+) ions, as shown in Fig. 6. The gate ions vibrate, harmonically, independently and perpendicularly to the path, with frequency xim (gate frequency) about stable positions, h, which are located symmetrically from the center of the path. h was estimated from the lattice parameters of each Pb1
xSnxF2. The allowed range within which a (Pb2+, Sn2+) ion vibrates is found in the Pb1
xSnxF2 system. The mobile ion also vibrates parallel to the path with frequency xm around a site from which it hops to the adjacent site. Both (Pb2+, Sn2+) and fluoride ions, having ionic radii rim and rm, respectively, are simply assumed to be three (Pb2+, Sn2+) ions moving independently. Now, we define the average overlap length d by d ¼ rm þ rim
h;

rim ¼ ð1
xÞrPb þ xrSn ;

respectively [26]. For d>0, the transition will occur only when the gate ions move away to give sufficient space for the mobile ion to pass through. The probability that the gate opens is given by ( ) n   R hþd=2
mx2im ðx
hÞ2     hþd dxexp 2kB T    ( )  ; ð9Þ Pn ¼  2  R hþd=2
mx2im ðx
hÞ    h
d=2 dxexp   2kB T where m is the average mass of the (Pb2+, Sn2+) ion, d is the displacement of the gate ion and n=1 assuming that three gate ions vibrate in phase (breathing mode). Eq. (9) for n=1 leads to

ð7Þ

where rm is ionic radius of fluoride ion and rm=0.131 nm for the coordination number (c.n.) of 4 and rim is given by ð8Þ

where rPb and rSn are ionic radii of Pb2+ and Sn2+ and 0.129 nm for c.n. of 8 nm and approximately 0.10 nm,

507

P1 ¼

    1
Erf fð2d=dÞQg    Erf ðQÞ


Q¼

2 mx2im d 2 8kB T

Erf ðQÞ ¼

2 p1=2

;

ð10Þ

1=2

Z

;

ð11Þ

dxexpð
x2 Þ:

ð12Þ

Q

0

The value of P1 depends on d and becomes constant with increasing the value of d. The constant value of P1 is used for analysis. The xp is described as
 D xp ¼ xim exp
ð13Þ P1 ; kB T

Fig. 6. Projection of fluoride ion and (Pb2+, Sn2+) ions on a plane.

where D is the difference between the potential minima for a mobile point particle at the center of the gate (saddle point) and a stable site [25]. In the hard-sphere model, the activation energy is determined only by the displacement of gate ions. Thus, we set D=0. The values of xp are obtained for Pb0.9Sn0.1F2 and Pb0.8Sn0.2F2 through the conductivity measurements, as shown in Table 1. On the other hand, xp is not obtained for h-PbF2 because the frequency-dependent ionic conduction process, which does not obey the power law of frequency, is not obtained as shown in Fig. 1a. Therefore, the activation energy Ea for the fluoride ion conduction cannot be estimated for h-PbF2. We assume that the values of the total activation energy E are equal to the values of

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S. Yoshikado et al. / Solid State Ionics 154 – 155 (2002) 503–509

Table 2 xp, xim, and overlap length d for Pb1
xSnxF2 solid solution system Sample

Lattice constant (nm)

Ea (eV)

xp0 (rad/s)

xim (rad/s)

d (nm)

h-PbF2 Pb0.9Sn0.1F2 Pb0.8Sn0.2F2

0.5934 0.5946 0.5957

0.442 0.304 0.257

0.6461013 1.221013 1.371013

4.321013 4.441013 5.201013

0.0148 0.0121 0.0097

Ea for all compositions because values of E of both Pb0.9Sn0.1F2 and Pb0.8Sn0.2F2 are almost equal to those of Ea, as shown in Table 1. P1 depends on both xim and the overlap length d (Eqs. (10) and (11)). Values of xim and d can be estimated using values of Ea and xp0 so that the temperature dependence of xp given by Eq. (13) agrees with that of measured xp. The estimated values of xim and d are shown in Table 2. Values of ionic radii of Pb2+ and Sn2+ estimated using the values of the lattice parameters and d are 0.126 and 0.104 nm for Pb0.9Sn0.1F2 and 0.126 and 0.105 nm for Pb0.8Sn0.2F2, respectively. These values are close to those (0.129 nm for Pb2+ and approximately 0.1 nm for Sn2+) reported in the literature [26]. On the other hand, the values of xp0 and xim of h-PbF2 estimated using values of Ea (Ea=E=0.442 eV) the lattice parameters, d, and the ionic radius of Pb2+ (0.126 nm) are 6.461012 and 4.321013 rad/s, respectively. The value of xim of h-PbF2 is close to that of Pb0.9Sn0.1F2 (4.441013 rad/s). Thus, it is suggested that the distribution of the activation energy, across which a mobile ion hops, is determined by the distribution of the overlap length d. It is suggested that the gate model is a simple model which explains the size-origin activation energy; it gives a consistent result with those of various Pb1
xSnxF2 systems, and the activation energy, across which fluoride ions hop to adjacent sites, is explained to originate from the overlap length d.

4. Conclusions Activation energies were calculated from the temperature dependence of the ac dynamic ionic conduction, and they could be separated into the activation energy for the hopping process and that for the generation of mobile carriers. The hopping frequency of fluoride ions could be estimated. (1) It became clear that all fluoride ions were mobile in the system of Pb0.9Sn0.1F2 and Pb0.8Sn0.2F2,

and each fluoride ion contributed to both the dc and ac ionic conductions. (2) The temperature dependence of the exponent of the power law for the frequency of the ac ionic conduction in the system of Pb 0.9 Sn 0.1 F 2 and Pb0.8Sn0.2F2 was weak (approximately 0.65) and showed the same frequency dependence as h-alumina and h-AgI. On the other hand, such conductivity was not clearly observed in h-PbF2. It was conjectured that the degree of disorder of the environment around the fluoride ion was small in h-PbF2. (3) It was suggested that the activation energy, across which fluoride ions hop to adjacent sites, originates from the overlap length d of (Pb2+, Sn2+) and fluoride ions.

Acknowledgements This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (no. 09215238) and in part by a High-Tech Research Center Project of Doshisha University.

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