Solid State lonics 5 (1981) 645-648 North-Holland Publishing Company

Testing the Assumptions of the Theoretical Calculations in Simple Superionic Conductors: The Phonon Response in KBiF 4

Gera ld Burns, F. H. Dacol, M. W. Sharer IBM Thomas J. Watson Research Cen te r

P. O. Box 218 York town Heights, New York 10598

G. D. M a h a n D e p a r t m e n t of Physics

Indiana Univers i ty Bloomington, Ind iana 47401

Theore t ica l calculat ions of the p h o n o n infrared and Raman response in simple superionic conductors such as AgI, CuI, and CaF 2 types are based on two fundamen ta l assumptions. First, most of the response can be unders tood in terms of a b r e a k d o w n of the select ion rules due to d isorder ( lack of t r ans la t iona l symmet ry) and second, ha rmon ic lat t ice dynamics can be used with a good degree of accuracy. This is tes ted here exper imental ly in the super ionic conduc to r K l _ x B i l + x F 4 + 2 x which has the CaF 2 s t ruc ture ( x = 0 . 0 is analogous to 2CaF2) and the assumpt ions are found to be appropr ia te . The calculat ions are also compared to recent exper imenta l results of the densi ty of s tates of PbF 2. Again agreement is good.

Two groups have done realist ic lat t ice dy- namic calculat ions on simple superonic conductors . A lben and Burns 1 have s tudied the wurtz i te phase of AgI a n d its high t empera tu re superionic conduc- tive phase, the a-phase . In the a -phase the I- ions are on the bcc latt ice points and the two Ag- ions per uni t cell are randomly dis t r ibuted over 12 possi- ble sites. They have also studied 2 the z incb lende phase of CuI and its h igh t empera tu re super ionic conduct ing phase, the aP-phase. In the a t -phase the I- ions are on the fcc lat t ice points and the four Cu- ions per uni t cell all are randomly dis t r ibuted over 8 possible sites. The Oxford group 3 (Elliott , Hayes, K leppmann , Rushwor th , and R y a n ) has s tudied CaF 2 and isostructural materials . In these mater ia ls there is no sharp phase bounda ry ; r a the r above some tempera ture , T c, the F- ions gradually start to have a non-negl ig ible probabi l i ty of occupying the large open sites in this crystal s t ruc ture and thus form F- in te r s t i t i a l ions. Bo th of these groups 1-3 have made two fundamen ta l assumpt ions in t rying to unde r s t and the p h o n o n spectra (about 10 c m - l ) . These are:

(1) The p h o n o n d i s t r ibu t ion can be under - s tood in terms of a crystal with a great deal of stat ic disorder (i.e. the a tomic mot ion associated with the dc conduct iv i ty is ignored) ;

(2) ha rmonic latt ice dynamics can be used as a good zero order approximat ion.

We have exper imenta l ly t es ted these two

assumpt ions using KBiF 4. This mater ia l has the CaF 2 crystal s t ructure 4 and is the x = 0 . 0 m e m b e r of the K l _ x B i l + x F 4 + 2 x system. As x increases the number of F- ion interst i t ials increases; this mimics CaF 2 in the superionic regime. Actually, the x = 0 . 0 mater ial is two phase; Sharer and C h a n d r a s h e k h a r 5 have found tha t only for 0 . 0 2 < x < 0 . 2 8 is the mate- rial single phase. Not only can the n u m b e r of F- ion interst i t ia ls be varied by varying x, bu t also at high t empera tu re (above 3 0 0 ° C ) the mater ia l is a super- ionic conduc tor . Thus, at low t empera tu re s the p h o n o n spectra can be studied for a d isordered ma- terial where there is no de conduct iv i ty ; at high t empera tu re the same mater ia l can be studied when it has a high conduct ivi ty . This differs f rom CaF 2

where the F- inters t i t ia ls occur only when there is dc conduct iv i ty due to the F - ion intersti t ials. Thus, the disorder and dc conduct iv i ty can not be separated.

The spect ra l m e a s u r e m e n t s tha t we have made are infrared reflect ivi ty and Raman measure- ments f rom two single crystals with d i f ferent values of x. R a m a n measurement s as a func t ion of temp- era ture ( 7 7 ° K to 4 0 0 ° C ) of these same two crystals have been measured. Last, R a m a n measurements , in this same tempera tu re range, of ceramic samples with the smallest and largest x values have b e e n measured. Figure l a and l d show the room temp- era ture results for the two crystals. Figures l b and lc show the results f rom a s tandard Kramers -Kronig analysis of the single crystal ref lect ivi ty data. The

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646 G. Burns et al. / Theoretical calculations in simple superionic conductors

0.6

~0.4

o.2 n ~

o) 0

3O

~ 2o

c~ 6 o") o,

4

2

c) 0

d)

ci t,l

e) 0

I I I I I I I I I

_ ~d~ . - I N \

, 1 I \

I I

i / / ~ ~ , R0.98Dq.02 r4.04 1 I" I I ,I I I I I I I I

100 200 500 400 500 600 ~(cm q)

Fig. 1 (a) Ref lect ivi ty at 2 3 ° C of two crystals of KBiF 4 wi th compos i t ions ind ica ted on the figure. (b) The conduct ivi ty , in units of (~2cm) -1 , ob ta ined

from the above reflectivi ty data using a Kramers- Kronig analysis (c) The energy loss, I m ( - e - I ) , of the above data also using a K-K analysis, where e is the dielectr ic cons tan t . (d) The reduced Raman data of the above crystals. (e) IRe d of two ceramic samples as labelled. The response at the high fre- quency end is due to f luorescence in our ceramic samples which we do not observe in our crystals.

L I I I I I I

1.0 / K0.95 Bil.O5 F4.10 ~,

i 0 .6

0 .4 t - , ~ ' / ~

.= / Z > !/\ 0 / I I I I I I

0 200 400 600 ENERGY (era -I)

Fig. 2 IRe d for an x=0 .5 ceramic at three di f ferent tempera tures as shown.

peak of the conduct ivi ty cor responds to the t rans- verse optic (TO) mode and the peak in the energy loss co r responds to the longi tudinal optic (LO) mode. The x var ia t ion of the TO mode is of inter- est and has been discussed previously. 6

For the purposes of this paper we can focus on the Raman results f rom the ceramics (Fig. l e ) . The results tha t are p lo t ted are the so-cal led re- duced R a m a n intensi ty, IRed, which are related to those directly measured, Is, by

~ls (1) IRed = 1 + n

where the Bose-Eins te in factor n = [exp ( h w / k B T ) - 1 ] -1 and hw is the energy of the phonon. IRe d ra ther than I s is normal ly displayed for wide spec- tral responses such as obse rved in glasses, lRe d involves only the densi ty of p h o n o n states and fre- quency dependen t matr ix elements , 7 e l iminat ing the well unders tood t empera tu re and f requency proper- ties of harmonic oscillators.

By measur ing the R a m a n spect ra be tween 77°K and 4 0 0 ° C we have found tha t IRe d is fairly i ndependen t of tempera ture , Fig. 2. This immedi- ately shows tha t the measured spectral response is most ly first order Raman indicat ing that ha rmonic lattice dynamics can be used as a good zero order approximat ion . This proves tha t the second as- sumpt ion is correct . Fur ther , the r e s u l t s found at very low tempera tures where there is no dc conduc-

G. Burns et al. / Theoretical calculations in simple superionic conductors 647

W

0

Z Ld O

i I I I

PbFz

0 I00 200 300

1 I I I I I I I

AgI WURTZITE CALC.

TA

AgI a - PHASE

CALC.

0 40 80 120 160 ENERGY (cm -I)

Fig. 3 Plots of the unnormal ized densi ty of s tates p (~) vs. energy. The results in t h e left are f rom exper imenta l measurement s in PbF 2 at two di f ferent tempera tures . 8 Those on the right are for AgI in two di f ferent s t ructures . l

tivity but there is static disorder are the s a m e results as those found at high t empera tu res where there is bo th static disorder as well as dc conduct ivi ty . This shows tha t the first assumpt ion is true, namely in this mater ia l the dc conduc t iv i ty has lit t le direct ef fec t on the p h o n o n spect ra , at least above

2 0 c m - 1 A last ind ica t ion of the appropr i a t eness of

the theoret ical calculat ion is given in Fig. 3. All of the plots show densi ty of latt ice v ibra t ional states, p(o~), in arbi t rary units, vs. energy. Calculat ions of o(0~) for two d i f fe ren t phases of AgI are on the right, 1 while exper imenta l values for PbF 2 are on the left. 8 The detai ls of the ca lcula t ions for AgI have been descr ibed previously. 1 In the o rde red wur tz i te phase the peak in o (~) due to the t r an - sverse acoust ic (TA) phonons , summed th roughou t the Bril louin zone, occurs at ~ 2 0 cm -1 and is la- beled TA. The smaller peak at -~70 cm -1 is due to the longi tudinal acoustic (LA) phonons . The optic modes occur above 80 cm - I and the high energy peaks are due to these modes. As can be seen in the high t empera tu re a-phase , there is still a peak in p (~ ) co r re spond ing to the TA p h o n o n branches , while at higher energies there is a smooth dis t r ibu- t ion in p(~) . (The same effect is seen in the calcula- t ions in CuI. 2)

The above calculat ions can be compared to exper imen ta l results of Dickens et al. 8 in PbF 2.

This material , which has the CaF 2 s tructure, is or- dered at low tempera tu res but is a f l u o r i n e super- ionic conduc to r at high t empera tu re s with a large number of F- ion intersti t ials. Since there are three atoms per primit ive uni t cell in the PbF 2 s tructure, the n u m b e r of p h o n o n b ranches t h roughou t the Bril louin zone is larger 9 than found in the wurtz i te structure. Thus, the 10°K, PbF 2 is more complicat- ed than found in the wurtz i te structure. The un- marked vertical lines at ~ 1 0 0 , 260, and 340 cm - I cor respond to zone center t ransverse optic, R a m a n active, and longi tudina l opt ic modes respect ively. In general they do not occur at not iceable posi t ions in p(~) . On the o ther hand, the peak at ~ 6 0 cm -1 cor responds to the TA modes summed through the Bri l louin zone. However , the L A mode peak at

140 cm -1 tends to be lost in a b road peak tha t also conta ins optic modes.

Now not ice the exper imenta l results for PbF 2 at 910 ° K. At this t empera tu re there are many interst i t ia l F- ions leading to a very disordered CaF 2 structure. The result shows one low energy peak ( ~ 4 5 cm - 1 ) which cor responds to the low tempera- ture TA modes but is shif ted to lower energy due to normal t empera tu re effects. The rest of the densi ty of s ta tes is just an extremely b road band. The simi- larity to the calculat ion for the a -phase AgI is strik- ing and lends s t rong suppor t to our belief tha t the calculat ions are basically correct .

648 G. Burns et al. / Theoretical calculations in simple superionic conductors

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3.

4.

References

R. Alben and G. Burns, Phys. Rev. BI6, 3746 (1977). G. Burns, R. Alben, F. H. Dacol and M. W. Shafer, Phys. Rev. B20, 638 (1979). See also Solid State Commun. 24, 753 (1977). R. J. Elliott, W. Hayes, W. G. Kleppmann, A. J. Rushworth and J. F. Ryan, Proc. Roy. Soc. London A360, 3t7 (1978). C. Lucat, P. Sorbe, J. Portier, J-M R6au, P. Hagenmuller and J. Grannec, Mat. Res. Bull. 12, 145 (1977).

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8.

9.

M. W. Shafer and G. V. Chandrashekha'r, this conference. G. Burns, F. H. Dacol, M. W. Sharer, G. D. Mahan, Sol. State Commun., to be published. R. Shuker and R. W. Gammon, Phys. Rev. Lett. 25, 222 (1970). M. H. Brodsky in "Topics in Applied Physics," edited by M. Cardona (Springer-Verlag, Berlin, 1975) p. 205. M. H. Dickens. M. T. Hutchings and J. B. Suck, Sol. State Commun. 34, 559 (1980). M. H. Dickens and M. T. Hutchings, J. Phys. C11,461 (1978).