Non-periodic local motion of silver ions in ßAg3SI from far-infrared conductivity measurements

Solid State Ionics 2 (1981) 341-346 North-Holland Publishing Company

NON-PERIODIC LOCAL MOTION OF SILVER IONS IN II-Ag3Sl FROM FAR-INFRARED CONDUCTIVITY MEASUREMENTS B. GRAS and K. FUNKE lnstitut fiir Physikalische Chemie und Elektrochemie der Universitiit Hannover, Hannover, West Germany

Received 28 September 1981

The frequency dependent far-infrared conductivity of B-Ag3SI, determined by measurement of transmittance and reflectivity, provides dynamic evidence for a non-periodic local motion of the silver ions. This motion takes place in the areas at the centers of the iodine-cube faces where the cationic potential is known to be flat. The experimental far-infrared conductivity is compared with conductivity spectra calculated on the basis of the following simple single-particle models: diffusion on a ring, statistical hopping between nearest-neighbor tetrahedral sites, and Brownian motion in a twodimensional harmonic potential.

I. INTRODUCTION In most ionic crystals the dynamics of the ions is sufficiently well described in terms of phonons plus jump diffusion. Even in many fast ionic conductors a reasonably well defined distinction can be made between the oscillatory and hopping motion of the ions. For example, this concept is still applicable to ~-RbAg415 (|). On the other hand, the local motion of ions is no more simply oscillatory, if the potential barriers between sites are sufficiently low, i.e., not exceeding the order of the thermal energy. This holds true in high-conductivity solid electrolytes like ~-Agl (1,2,3). In these cases the motion of the mobile ions should be more suitably described by stochastic models using continuous periodic potentials (2,3). Unfortunately, however, Systems like ~-Agl are far too complicated for any stochastic single-particle approach, since the motion of the particles is highly correlated. Hence the quantitative study of non-oscillatory movements of ions in solids calls for dynamically simpler systems. They should be characterized by I) shallow potentials, producing non-periodicity of the local ionic motion, 2) approximate constancy of these potentials as functions of time in order to ensure the applicability of simple single-particle models, and 3) almost stochastic driving forces. Evidently, the second property is equivalent to "dynamic simplicity" in the sense of negligible importance of correlations between the movements

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of different

particles.

2. THE SYSTEM B-Ag3SI In the following we will show that the above requirements are well met by the solid electrolyte B-Ag3SI. The B-phase of Ag3SI, stable from 157 K to 519 K (4,5,6), is simple cubic (space group Pm3m) with a lattice constant of 4.897 ~ at 295 K and may be regarded as a modified antiperovskite structure (5,7,8,9). The I- and S 2- ions are on the 0,0,0 and I/2,1/2,1/2 sites, respectively. For the sites of the silver ions, Reuter and Hardel proposed the largest voids available in the anion structure, situated at the 12 fold tetrahedral position 0.4,1/2,0 [12(h)~, only 0.5 A apart from the face center (7,9). There is a group of four of these sites in the vicinity of each face center, and the number of face centers coincides with the number of silver ions. Normally, for geometrical and electrostatic reasons, only one site out of a group of four will be occupied at a time, resuiting in a confinement of each silver ion to its cube face and in a relatively low electrical conductivity. (The conductivity at 295 K is by two orders of magnitude lower than in the ~-Agl type s-phase at 523 K, where the anions are statistically distributed over the sites of a bodycentered cubic lattice (8,9,10).) In Fig. I the location of the tetrahedral sites is compared with the electron density distribution due to the silver ion. The Fourier map

B. Gras, K. Funke ~Non-periodic local motion o f silver ions in #-Ag3S1

342

reproduced in the figure has more recently been obtained by Perenthaler et al. from single-crystal X-ray data (7). The authors point out that two different structure models are likewise consistent with the Fourier map. The first is based on Reuter's and Hardel's model, with a distribution of the silver ions over the tetrahedral sites. In the other, the octahedral site I/2, I/2,0 is assumed to be occupied with a probability of one. As should be expected from Fig. I, unusually large temperature factors are obtained in both refinements.

~

} 11

I

~.' ;

I

/

\\ i

,, \ ,J ,

,

Fig. I. Fourier map of the z=O section of B-Ag3SI , according to Perenthaler et al. (7). Contours start at I e ~-3 and are at intervals of 2 e ~-3. For comparison, the tetrahedral sites are marked by dots.

The results of Perenthaler et al. provide structural evidence for a shallow wide shape of the potential energy surface throughout an area comprising both the tetrahedral sites and the octahedral site. This means that B-Ag3SI fulfils our first requirement. The path of an individual silver ion in B-Ag3SI is determined by the action of the following forces. Besides the gradients of the local potentials caused by the presence of the nearest neighbors, forces essentially arise from the coupling to the phonon system and from the electrostatic interaction with the moving charges of the other silver ions in the crystal. These forces can be considered practically stochastic, since they vary fast compared to the time the ion needs to traverse the local area available for it. On the other hand, the latter time is small compared to the time the ion stays

within this area, as can be seen from the value of the electrical conductivity. For an approximate description of the local dynamics, translational hops from a face-centered area to another may therefore be neglected because of their relatively rare occurrence, in spite of their high.\ ly correlated character. In this approxlmatlon the silver ions are thus considered as particles moving in areas with invariable, but structured, flat potentials, driven by stochastic forces. This means that our requirements 2 and 3 are also met by 8-Ag3SI. At this stage two consequences have to be pointed out. (i) There should be a contribution to the frequency dependent very-far-infrared conductivity, o(~), due to the non-periodic motion of the silver ions within the areas on the cube faces. Indeed, Br~esch and Beyeler have already reported a continuous increase of the far-infrared reflectivity at wavenumbers below 40 cm -I, corresponding to a maximum of ~ below 40 cm -I (11,12). (ii) The very-far-infrared contribution to o(v) can approximately be explained in terms of the single-particle motion of the silver ions. Thus simple dynamic models can be tested against experiment by comparing calculated single-particle conductivities, Olp(~) , and the experimental conductivity, o(~)i In this paper we present the far-infrared conductivity spectrum, o(~), of B-Ag3SI at 295 K as obtained from spectra of both reflectivity, R(~), and transmittance, T(~). For comparison we also present single-particle spectra, Olp(~), derived from the following simple models: A) diffusion on a ring, B) statistical hopping between tetrahedral sites, C) Brownian motion in a two-dimensional harmonic potential. .

3. EXPERIMENTAL Silver sulfide iodide was formed by adding a solution of AgNO3(P.a., Merck) dropwise into a solution of Na2S and KI (p.a., Riedel-De Haen) at 70 °C. The product was thoroughly washed with conductivity water. Afterwards, it was dried in a vacuum for several days at a temperature which was slowly increased to 200 °C. The positions and intensities of X-ray reflections from the material were found identical to those reported by Reuter and Hardel (9). In the experiments, polycrystalline pressed pellets have been used. Photochemical decomposition of the samples was carefully avoided. Spectra of transmittance, T, and reflectivity, R, were taken at 295 K using Beckman Fourier spectrophotometers FS 720 and LR 100. The latter instrument is equipped with a lamellar grating instead of a Michelson interferometer and is preferable at frequencies below I THz. The thickness of the samples was chosen to be larger than 2 mm and typically 0.1 rm~ for the reflectivity and transmittance measurements, respectively. In reflection geometry, the reflection angle was 12 ° .

B. Gras, K. Funke /Non-periodiclocal motion of silver ions in ~-Ag3S1

4. RESULTS The spectra of reflectivity, R, and transmittance, T, obtained from B-Ag3SI at 295 K are presented in Fig. 2.

343

ent ways. First, the refractive index, ~(v), the permittivity, ~(v), and the electrical conductivity, o(~), are determined via the equations

R =

I~12

(la)

r = l(1+r).exp(-ind~/c)-(1-r)l 2 . . . .

2O

\

I

~

100

5O

(cm -1 } 200

0.2

[ I

1

= ~2

(Id)

= iSo~

(le)

o = Reo

0.3

0.5

(Ic)

500

0.4

0.1

(Ib)

= (1-n)/(1+n)

_./', k 5

2 ----=,,,-

J 10

V (THz)

Fig. 2. Experimental reflectivity and transmittance spectra of B-Ag3SI at 295 K.

In its B-phase Ag3SI is essentially opaque in the entire frequency range from 0.5 to 10 THz. The reflectivity displays significant structure corresponding to three maxima of the frequency dependent conductivity, near 0.5, 2.5, and 8 THz. In particular, there is an increase of R below I THz as reported by BrOesch and Beyeler (11, 12). This increase of R and the matching low values of T at frequencies below I THz are not found in the low temperature phase, y-Ag3Sl , where the silver ions occupy well defined sites. It is therefore concluded that the features displayed by 6-Ag3SI in the very far infrared are due to the local non-perlodic motion of the silver ions in this material. The frequency dependent conductivity, o(v), is calculated from T(v) and R(v) in two differ-

(If)

Here, ^ denotes complex quantities; r, d, ~, and c are the reflection factor, the thickness of the sample, the angular frequency, and the velocity of light in a vacuum, respectively. In Equation (Ib), multiple-reflection effects are neglected because of the high absorption in the sample. In fact, no oscillations caused by multiple reflections can be detected in the spectra. Second, a Kramers-Kronig analysis of the reflectivity data yields permittivity and conductivity. The conductivity spectrum of Fig. 3 is obtained by combined application of both techniques. In particular, the Kramers~Kronig analysis is indispensable for determining the heights of the conductivity maxima at 2.5 and 8 THz. On the basis of our present experiment on polycrystalline samples, these conductivity maxima are not readily assigned to specified TO lattice modes; to do so, one would need single-crystal inelastic neutron scattering spectra and/or lattice dynamical calculations. In the following we will entirely concentrate on the broad low-freq~ency conductivity maximum and compare it to the predictions from simple single-particle models for the local non-periodic motion of the silver ions.

5. COMPARISON WITH MODELS Our single-particle models, i.e., diffusion on a ring, statistical hopping between nearestneighbor tetrahedral sites, and Brownian motion in a two-dimensional harmonic potential, constitute three extreme and complementary ways how to conceive the Ag + motion within the area at the center of Fig. I. A) Diffusion on a ring The diffusion equation on a ring,

~2 D.7#

G~(,,t) = ~ G~(*,t)

(2)

R with boundary conditions Gs(~,O) = 6 ( ~ ) a n d ~G~(~,t)/3~ = O, is solved by

G~(~,t) = ~-~_~ I ~ cos(m~).e-DRm2t

(3)

Here, ~, t, DR, and G~ denote angle, time, rotational diffusion coefficient, and the selfcorrelation function for rotational diffusion,

B. Gras, K. Funke ~Non-periodic local motion of silver ions in ~.Ag3SI

344

5

10

20

50

100

O(cm -1 ) 200

r ,

/

J

i

o

500

(Q-lcm-1)

i ooo; ~ I

0

xeXIXIX e

0

X@X@

.l.°.x.X"

0 ex

,: l'

Ol

0.2

05

1

\

(i

ex

0

aX

O

O

2

10 v (THz)

Fig. 3. Electrical conductivity of 6-Ag3SI in the far infrared at 295 K. Solid line: experimental data; crosses: model A; dots: model B; open circles: model C.

R

respectively. From G s ehe mean square displacement and the velocity autocorrelation function are obtained: <(r(t)-r(O))2>E = ~ 4R2(1-cos~)g~(~,t)d~ 0 = 2R2"(1-e -DRt) (4a) <~(0)'~(t)> R

= R2DR'6(t)-R2D~'e-DRt.

(4b)

R is the radius of the ring. So far, however, Equations (4a) and (4b) only hold for macroscopic applications. A Langevin-type short-time behavior is therefore introduced: kT b = m.DR.R 2

(5)

R kT .{(b-DR)e-(b-DR)t (6) --= m(b-2DR) _

DRe-DRt}

In Equations (5) and (6), m, k, T, and b denote the mass of the diffusing particle, Botzmann's constant, temperature, and the friction term, respectively. Fourier transformation of the velocity autocorrelation function yields the frequency spectrum and the singleEparticle contribution to the conductivity, alp(~) : K q~p(~) = ne 2 • b2~ 2 3mb{(bDR_D~_ 2)2+b2 2}

(7)

Here, n is the number density of the mobile par ticles and e is their charge. The function a~n(~)2attains'2y its maximum value at w = (bDR-DR)I/ and is symmetric on a log~rithmic frequency scale. In order to apply the ring-diffusion model to 6-Ag3SI, we first determine R by equating the area of the ring and the area of the square defined by the tetrahedral sites. The result, R = 0.4 ~, defines b'D R by Equation (5) and implies a value of ~o slightly below (b'DR) I/2 = 2~.0.6 THz. Choosing ~o/2~ = 0.5 THz, in agreement with the experimental conductivity spectrum, one obtains the q~p(~)_ spectrum plotted in Fig. 3. On the basis of dynamical arguments, one would expect the friction term b to be of the order of a few reciprocal picoseconds. In our example, the numerical value of b is 6.8 ps -I, while D R is 2.1 ps -I. It is noted that the maximum of O~p.iS_ lower than the experimental conductivity maxzmum near 0.5 THz. B) Statistical hopping between nearest-neighbor tetrahedral sites Any preference of the silver ions for tetrahedral sites was ignored in model A; on the other hand, we exaggerate the importance of these sites in model B. In the simplest possible hopping model, statistical hops between nearest-neighbor tetrahedral sites are considered, and it is assumed that the time of flight, T I, is small compared to the mean residence time, t o. The latter is

B. Gras, K. Funke / Non.periodic local motion of silver ions in frAg3$1 defined by the assumption that the probability of finding a particle on a site where it was at t=0 decreases as exp(-t/To). This Debye-relaxation type model predicts an increase of the conductivity as a function of frequency, proportional to (~0To)2/(1+(00To)2). By comparison with the experimental conductivity dispersion, T o is found to be about I ps which implies that the assumption T I << T o cannot hold true. We therefore incorporate a fixed non-zero time of flight into the model. The treatment via the velocity autocorrelation function is straightforward and results in single-particle conductivity spectra O~p(~0)_ which now decrease on their high-frequency side. The Debye-relaxation solution is recovered in the limit of TI<
2 2 2 ne "~o.a

.~(m)

(8)

6kT.'r21 • (To+T 1) ~(C0) = 2 x - 2 ( l - c o s y ) + x z ( 2 s i n y - s i n 2 y ) + z(l-2y÷cos2y) x=~0To ; y=~0TI ; Z=(X2+X4) -I In Equation (8) a is the distance between nearest-neighbor tetrahedral sites. The particular spectrum oH_(~0) shown in Fig. 3 is based on a = ~ - 0 . 5 ~, T o = O ~ ps, TI=0.35 ps and has its maximum position at 0.5 THz. Evidently the difference between models A and B, concerning the fine structure of the non-periodic motion on the ring, has practically no effect on the shape of the respective singleparticle conductivity spectra; it is therefore impossible to decide in favor of one of them from the experimental data. It is noted that., the heights of the maxima of both oR~(0~)j and O~p(~0). are smaller than the conductiv1~y observed near 0.5 THz. Some feature of the actual motion of the silver ions is obviously not yet taken care of by models A and B. C) Brownian motion in a two-dimensional harmonic potential The motion of a Brownian particle in a twodimensional harmonic potential is described by the Langevin equation mr + mbr + m0~ r = --o--

F . -stochast ~c

(9)

The velocity autocorrelation function and the frequency spectrum derived from Equation (9) have the same algebraic structure as those from model A. The single-particle conductivity, O~p(~) = 2ne2"b2w2 2 2 3mb.{(~o-~ )2+b2w2} R

,

(10)

differs from oi_(~) by a factor of two. This is due to the fact~that R=kT/m is replaced by B=2kT/m_ in the two-dimensional treatment. The parameter values used for the calculated spectrum O~p(~). plotted in Fig. 3 are ~o=2~'0.5

345

-1 THz and b=6.8 ps , as in the case of the ringdiffusion model. This corresponds to overdamped, non-periodic motion, since b>2~ o. The maximum height of o~ (~) is now slightly above the experimental v~lue.l In contrast to models A and B, model C yields a bell-shaped Ag+probability density, centered at the octahedral site. Its width may be characterized by =(2kT/~m) I / 2 ~ I and by I/2= (kT/m) I/2~1.--Insertion qf ~o=2~-0.5 THz--yields = 0.38 ~ and <~2>I/Z = 0.48 ~. These values do not seem to contradict the structural information contained in the Fourier map of Fig. I.

6. CONCLUSION The very-far-infrared conductivity of B-Ag3SI provides dynamic evidence for a non-perlodic local motion of the silver ions in this material. Comparison with three calculated single-particle spectra has shown that, with regard to position, shape, and width, all of these models are consistent with the experimental data. They are also consistent with the Fourier map of Fig. I. A further specification of the shape of the actual cationic potential thus requires consideration of the height of the 0.5 THz conductivity maximum. To this end it has to be assumed that, in this frequency range, the ratio ~/Olp is sufficiently close to unity; as argued in-part 2, B-Ag3SI is a very good candidate for this assumption. The inequalities (Olp)ma x~ H

( ° F ) m a x < (oeXP)ma x < ( ~ - ) m a x P

(11)

hence indicate an actual cationic potential intermediate between those of models A and B, and model C. The most probable potential is thus characterized by a double-minlmum cross-section and approximately rotational symmetry in the z=0 plane. Particularly low values of the potential energy are of course expected at the tetrahedral sites. A stochastic single-particle calculation based on this kind of potential will most probably give rea1{stic results, but is not easy to perform. Also, it would contain more parameter values than can be fitted to experiment. Schneider and Str~ssler have put forward a stochastic model with a double-well potential between tetrahedral sites (13). Of course this model has the shortcoming of being one-dimensional, like models A and B. Brdesch and Beyeler have estimated a barrier height of k.350 K between nearest-neighbor tetrahedral sites from their far-infrared reflectivity spectrum (11). However, in view of the close similarity of the conductivity spectra obtained from models A and B, the determination of any significant barrier height between tetrahedral sites from far-infrared data appears at least questionable.

346

B. Gras, K. Funke ~Non-periodic local motion o f silver ions in #-Ag3S1

ACKNOWLEDGEMENT We wish to thank Messrs. H. Witek and J. Gast who confirmed the validity of the high-frequency part of our reflectivity spectrum using Digilab and Bruker Fourier spectrophotometers. We also thank Dr. H. Schulz for sending us his Fourier map. The major parts of our instrumentation were kindly supplied by the Stiftung Volkswagenwerk. Financial help from Fonds der Chemischen Industrie is gratefully acknowledged.

REFERENCES (I) K. Funke, Advances in Solid State Physics 20 (1980) 1 (2) P. Fulde, L. Pietronero, W.R. Schneider and S. Strgssler, Phys. Rev. Lett. 35 (1975) 1776 (3) T. Geisel, in: Physics of Superionic Conductors, Ed. M.B. Salamon (Springer, Berlin,

1979) p. 201 (4) S. Hoshino, T. Sakuma and Y. Fujii, J. Phys. Soc. Japan 45 (1978) 705 (5) S. Hoshino, T. Sakuma and Y. Fujii, J. Phys. Soc. Japan 47 (1979) 1252 (6) A. Magistris, G. Chiodelli and A. Schiraldi, Z. Phys. Chem. 112 (1978) 251 (7) E. Perenthaler, H. Schulz and H.U. Beyeler, Acta Cryst. B 37 (1981) 1017 (8) B. Reuter and K. Hardel, Naturwissenschaften 48 (1961) 161 (9) B. Reuter and K. Hardel, Z. Anorg. Allg. Chem. 340 (1965) 168 (10) B. Reuter and K. Hardel, Ber. Bunsenges. Phys. Chem. 70 (1966) 82 (11) P. BrHesch and H.U. Beyeler, Helv. Phys. Acta 50 (1977) 593 (12) H.U. Beyeler and P. BrNesch, Bull. Am. Phys. Soc. 22 (1977) 369 (13) W.R. Schneider and S. Strgssler, Z. Physik B 27 (1977) 357