Low-frequency, coulomb-correlated density fluctuations in superionic conductors and their influence on the conductivity

~I~’)Solid State Communications, Vol.36, pp.33—37. ‘~“Pergamon Press Ltd. 1980. Printed in Great Britain.

LOW—FREQUENCY,

COULOMB-CORRELATED DENSITY FLUCTUATIONS IN SUPERIONIC AND THEIR INFLUENCE ON THE CONDUCTIVITY Roland Zeyher

Max—Planck-Institut Received

für Festkorperforschung, Fed. Rep. Germany July

1,

1980,

by Manuel

7 Stuttgart

CONDUCTORS

80,

Cardona

The low—frequency density and current response of diffusing ions is calculated for superionic conductors. In particular, it is shown that the peculiar dispersion of the conductivity of 0-AgI at low frequencies is due to the scattering of silver ions by slowly relaxing local density fluctuations caused by the other silver ions.

The ionic conductivity 0(w) of the superionic conductors ü—AgI, ~—CuBr, and 0—Cul behaves in a 1rather peculiar way at low frequencies : with increasing frequency, 0(w) first decreases by about a factor two a?d then shows a broad peak at about 1 cm . Such an w de~endence can be qualitatively obtained in a single particle hopping model assuming that the potential barriers are modulated by weakly daTped lattice modes with frequency ‘~1 cm . The purpose of this letter is to present an alternative ex— planation according to which the dispersive part of 0(w) at low frequencies is due to the interaction of the current with slowly varying, highly correlated fluctuations in the density of diffusing ions, The microscopic equations for the various space distribution functions can be greatly simplified if one takes advantage of the following two inequalities: << w (1) 2/(Ca) kBT <<

mal

Inequality (2) states that the ther— energy kBT is much smaller than the

average potential energy between diffu— sing ions (Z is the static ionic charge, C a background dielectric constant, a the lattice constant) . This inequality is very well fulfilled in the above superjonic con— ductors. As5shown by a more detailed investigation this inequality justifies the use of only the lowest powers in the momenta instead of the microscopic distribution function in momentum space. Inequalities (1) and (2) are most easily taken into account by restricting the space ~ of all dyn~mical variables to a suitable subspace is spanned by the coarse grained density and momenturn density variables for the diffusing ions:

~X

J 1

~

=

p~(kp)

—1/2 N

(2)

(Ze)

1 1,0

exp(ikx(1U)L(

(3)

is the hopping frequency for diffusing ions and has been estimated by sirn?le one-parti~le models to be about 2.5 cm for ~—AgI . w is the frequency of the lowest critical point in the phO?on density of states ~nd is about 2o cm in the case of a-AgI . Inequality (1) means that the motions of the ions within one potential well (for instance oscillations around potential minima) are much higher in frequency than the jumps of the ions between different potential wells. Restric— ting ourselves to frequencies w<
33

and their product states which are at most linear in the momenta. The index 1 labels the N primitive cells of the crystal. x(a) is the position vector ofthe diffusing particle 0. Each ion is exposed to the periodic potential produced by the nondiffusing ions. The equivalent absolute minima of this potential within the pri‘nitive cell 1 are located at x(lll). For instance in the case of m—AgI there are one silver ion per primitive cell and six equivalent potent~a~ minima at the tetra— hedral sites, a (r—x(1l.I)) is Wannier function locates at the site x(IU) which can be formed from the lowest Bloch-like eigenfunctions of the underlying Smolu— chowski equation. Furthermore let us define a scalar product (A~B) of two dyna-

34

COULOMB—CORRELATED DENSITY FLUCTUATIONS IN SUPERIONIC CONDUCTORS

mical variables A,B by /(k T) where is the conjugate complex of ABand < > denotes the average over the canonical ensemble. We can then define the pr~jec— tion operator P onto the subspace and the corresponding projected Liouville operator~ ,~=‘~‘~P where~is the Lieuvilla operator. The correlation functions a~sociated with the density variables n(k~i) can be calculated from K~bos relaxation functions defined as

~

-1

~‘

~(i~i

1p2,z)=(n(ku1)I(z—~)

According to ~ori wing equation :

-~

~ satisfies

the

~[z6~~

—~

M(k~1~4,z)X

follo-

~

1(k~

Some of the eigenvalues of k B TX are rather large ( %25 near the zone boundaries) . These large values signa— lize an incipient instability of the homogenous distribution of Ag ions towards formation of a superstructure by means of static charge density waves. Phase transitions of this kind are expected because the average potential between the ions is large compared to their thermal energies. The very delicate balance between short- and longrange correlations in X can be seen froc the relative importance of its various contributions. For instance in the case

of the

n(kU2))

4p3)

Vol. 36, No.

eigenvalue

of

X

(k=o,U1U

the ther~l contribution (~ o.233 eV? is small compared to the Coulomb contribution (~—o.995 eV) as well as the hard core contribution (“~o.842 eV) . Hoewever the latter two cancel each other to a large which

extent leading to a total value is of the same order as the ther—

3

~(kU3U2,z)

with

the

static

X(kU1(~2)

=

susceptibility +

x(ku1~~2) =

(5)

matrix

+

(n(kl~1) In(k~2))

(6)

mel contribution. Near the zone boundaries the cancellation is less perfect in some cases giving rise to large values of X~ The following two approximations were made in evaluating M: a) The Fourier transformed kernel M(l1U1,l )12,z) was taken to be short-ranged so tha~ only self—terms and nearest neighbor terms are nonzero. The resulting two different matrix elements are connected by charge conservation. It is thus sufficient to calculate, m(z) defined by

and the mornory mgtrix M defined for instance in Ref. . Thus the calculation of ~ is reduced to the calculation of function) and of the M. pair correlaXtion (i.e., essentially In the mean spherical approximation X satisfies the equation

2 M(kp1p2,z)

a(w) ~

(sk To B

for

is

=

instance,

+ 0(k4);

(8)

m(z)k

then proportional

to

—In m(w—ifl)

+ C(ku -~

1u3))x(k~3~2) -~

was Using expressed the forceb) again in theterms Mon oftechnique m(z)

1•13

force—correlation =

5

function

F(z)

(7) m(z)

The factor s is equal to the number of eq~ivalent minima per primitive cell, C(k1J1112) is the Fourier transform of the direct correlation function C(r) where the argument r is restricted to all possible distances between the potential minima. In the case of m—AgI2C(r) is equal to the RPA value -(Ze) /(CrkBT) for r>d (d is the distance between nearest neighbor tetrahedral sites). The remaining two short—range values C(r=o) and C(r=d) are determined from the condition that the pair correlation function vanish for these distances due to hard core effects. Using Z=l, c=9, T450K we obtai~ed1rtumerically the values k TC(r)e A =-o.o59 for r=o and —o.o56 f~r r=d, respectively.

with

=

_(j~Ij0)/(z+F(z)/(i~j0))

the projected

=

(9)

current

S/M~~(op)

(lo)

is a Cartesian index and M the mass. S denotes a nearest neighbor overlap matrix element of the first derivative of the function a (see Eq. (3)) with another function a. In our application to a—~gi S was taken to be o.o24 which is not inconsistent with a sum rule ar— gumen~ assumin~ that only the dispersive part 1.5 cm in o(w) is due to j F is calculated in the mode—mode coup~ing U

1

Vol.

36, No. 1

35

COULOMB—CORRELATED DENSITY FLUCTUATIONS IN SUPERIONIC CONDUCTORS

8 approximation assuming that the current decays into product states of density

diffusion coefficient. The presence of Coulomb forces causes a finite relaxation +

fluctuations

via the

and hard core

screened

Coulomb

frequency 4iT~(o)/C for ment with hydrodynamics are two modes of I”~2 andl’

interactions:

;

25

T(~~~p~) T(qu3~4) ~ q 1’

dw1dw2

J

(z-w1-w2)~

~

÷

II

+

(-q~1~3,w1)~(q~2~4,w2)

with These tuations relaxation modes where describe the frequencies Ag local ions density are “~2.5 redistricm fluc—

of such nodes in superionic conductors (Refs. 11—13) has previ~~sly been discussed in terms of spin3 j~r single particle hopping models ‘ . The experimental value of 3.6 cm’for the mode is in rough agree~1ent with our th~reti-

(11)

is the spectral function of ~. T denotes an effective vertex containing essentially the Static susceptibility and S. Eqs. (5), (8), (9), and (11) form a closed s~t of equations for c~ and m which must be solved self—consistently. At the beginning of the first iteration step m(z) is approximated by its zero frequency value. The resulting density fluctuations are purely diffusive and decay exponentially in time. The calculated relaxation frequencies (imaginary part of the eigenfrequencies) are÷shown in Fig. 1 for U—AgI at T4S0K. At k=o there are two kind of density fluctuations: a)+There is a very fast re— laxing mode of r~1symmetry (relaxing frequency ~17 cm ) where the density fluctuates on a macroscopic scale. With— out Coulomb forces this mode would be a hydrodynamic diffusio~ mode with a relaxation frequency Dk where D is the

cal value 2.5 cm . Near the zone boundaries the relaxation of some of the modes is drastically slowed down, reflecting the incipient instability to— wards formation of static density waves and the high correlation of the ions due to Coulomb and hard core interactions. The first iteration is concluded by calculating F via Eq.(11) using the above dispersion curves for density fluctua— tions. The result for o(w) is shown as dashed line in Fig. 2. Most remarkable is that o(w) decreases with increasing freeque~cy with a half-width of about o.7 cm reflecting the density of product states of density fluctuations. The reasonable agreement of the calculated value 0(o) (which is independent of the special choice for S) with experiment shows that the considered scattering pro— cess is indeed the dominant one at low frequencies. In the second iteration ~ is recal— culated from Eq.(5) using the m(z) of the first iteration. Since m v~ries substan= tially on a scale of 1 cm the density

15

E 3 010 ‘II-

a 0. a

.~

5

0)

a

0

r

‘i..

[lçK,o]

______

N p

______

[tçiçKJ r r

[K~o~o]H

Wave Vector Fig.

1:

symmetr~1

buted among the six tetrahedral sites within the primitive cell. The presence

114

II

agreeb) There

Dispersion of the relaxation frequencies for density fluctuations in U-AgI at T450K using the low—frequency limit of m(z).

COULOMB—CORRELATED DENSITY FLUCTUATIONS IN SUPERIONIC CONDUCTORS

36

2.5-

—

-I-, ~ 2.0 -\

Vol.

36, No. 1

15-

,~10•

Exp.

~‘

Wave number v[cm~J Fig.

2:

Theoretical conductivities for U—AgI at T=45oK after the first iteration (broken line) and the second iteration (solid line). Inset: experimental c 9nductivi— ty of 0-AgI at T=523K

fluctuations are no longer purely di— ffusive (due to the imaginary part of m) but also have oscillatory features (due to the real part of m) . The solid line in Fig. 2 shows the recalculated con— ductivity using Eqs.(9) and (11) and the new expression for ~ (in order to per— form the w integrations analytically we have approximated the dashed line in Fig. 2 by one Lorentzian) . 0 is no longer monotonically decreasing with w but ex:1 hibits a broad peak between 2 and 3 cm in semiquantitative agreement with th~ experimental curve shown in the inset . The origin of this peak may be understood in the foi~owing way: for fre— quencies >>1 cm the high—frequency approximation F(z) = -c/z applies rough1’~~2.8cm , Thefinds ly. Inserting this into Eq. (9)1one 0(w)’~5(w5—function, however, with ‘f appears in our more realistic calculation only as a broad peak in the spectrum due to Lan,~,au damping because the frequency ~?jc does not represent an isolated pole of m(z). After the completion of this manuscript new experimental data for 0(w)

‘f~’)

of 0—AgI for frequencies b~ween 4 and 40 GHz have been published . According to these authors 0 is rather constant or even increases slightly with fre— quency in this region. These data disagree with those of Ref. 1 and also with our theoretical results. We would ~ke to point out that the new data of are also inconsistent with Ref. 16: At w = 40 GHz the a~,sorption constant U at T~523K is 80 cm . Taking the refractive index ‘{~T of Ref. 15 and interpolating o(w) from T=523K to T429K according to the known temperature de—1 pendence of 0(o) o(w ) is o.68 (0cm) at T429K in contrast°to the value 1.2 of Ref. 15. Also the infrared data for w>2oo GHz of Ref.1, which have been that the flat region a between 40 confirmed by Ref. 17, instrongly indicate and 90 GHz of Ref. 16 lies substantially below the static value 0(0), in contrast to the results of Ref. 15. AcknowledgementIt is a pleasure to thank W.G. Kleppmann for many discussions and P.S. Allen and A. Martin for comments on the manuscript.

REFERENCES 1. 2.

K. Funke, Progr. Sol. St. Chem. 11, 345(1976) L. Pietronero, S. StrAssler, and H.R. Zeller, Sal. State Comm. 3o, 3o5(1979)

3. 4.

W. Dieterich, T. Geisel, I. Peschel, Z. Physik B29,5(1978) N. Bührer, P. BrUesch, and S. StrAs StrAssler, Proc. Sec. lot. Meeting on Solid Electrolytes, St. Andrews,

Vol.

36, No. I

COULOMBCORRELATED DENSITY FLUCTUATIONS IN SUPERIONIC CONDUCTORS

(1978), to appear in Electrochim. Acta 5. W. Kleppmann and P. Zeyher, to be publ. 6. D. Forster, Hydrodynamic Fluctua— tions, Broken Symmetry, and Corre— lation Functions, W.A. Benjamin, London (1975) 7, E, Waisman and J.L. Lebowitz, J. Chem. Phys. 52,43o7(197o) 8. W. Goetze and M. LOcke, Phys. Rev. A11,2173(1975) 9. R. Zeyher, Z. Physik B31,127(1978) lo. J. Jãckle, Z. Physik B3o,255(1978) 11. R’~A. Field, D.A. Gallagher, and M.V. Klein, Phys. Rev. B18,2995 (1978) 12. G. Winterling, N. Senn, M. Grimsditch, R, Katiyar, Proc. mt. Conf.

13.

14. 15.

16. 17.

Lattice Dynamics, ed. by M. Balkanski, Flammarion, Paris(1978), p.553 R.J. Nemanich, R.M. Martin, and J.C. Mikkelsen, Sal. State Comm. 32,79(1979) B.A. Huberman and P.M. Martin, Phys. Rev. B13,1498(1976) K.F. Gebhardt, P.D. Soper, J. Mers— ki, T.J. Salle, and W.H. Flygare, J. Chem. Phys. 72,272(198o) H.R. Chandrashekar and L. Genzel, Z. f. Physik B35,211(1979) H.R. Zeller, P. BrQesch, L. Pietro— nero, and S. StrZssler, in Superionic Conductors, ed. by G.D. Mahan and W.L. Roth, Plenum Press, New York (1976), p. 2o1

37