PH YSICAL REVIEW B VOLUME 22, NUMBER 2 15 JU L Y 1980

Structure and dynamics of superionic conductors. II. Collective excitations andsingle-particle motion

W. Schommers

(Received 27 December 1979)

Molecular-dynamics calculations have been performed for a two-dimensional disordered many-particle

system with strong anharmonicities which is embedded in a periodic potential. The disordered system is

formed by 128 silver ions and the periodic potential is given by a well-defined iodine lattice also consistingof 128 ions. Although restricted to a two-dimensional model, the results for the main dynamical propertiesof the system agree well with various experimental data for a-AgI. Scattering functions have been used in

order to study the particle motion. It turned out that the shape of the coherent scattering law can be

interpreted in terms of jump-diffusion processes. Furthermore, we have found that the individual motion is

strongly coupled to the collective modes (in particular optical modes) of the system.

I. INTRODUCTIO'N

Superionic conductors are solids with high con-ductivity comparable to liquid electrolytes. Today,the principal lines of development are (i) thesystematic search for new solid electrolytes (e.g. ,in view of the construction of solid state batteries)which exhibit high ionic conductivities and highenergy densities at ambient temperature, and (ii)the investigation of the microscopic mechanismof superionicity in these materials. It seemsprobable that a good understanding of the micro-scopic mechanism might help to select new solidelectrolytes. Whereas the application of superionicconductors to batteries, etc. is partly in an ad-vanced stage, the microscopic understanding ofthe structure and the dynamics of these materialsis just at the beginning. This is the reason why themost theoretical investigations are still concen-trated on simple systems, in particular, the so-called AgI-type solid electrolytes.

The ions of a solid electrolyte can be dividedinto two groups (called A type and B type in thefollowing). The A-type ions form a disorderedsubsystem and show strongly anhaxmonic behavior.This is due to the extremely high mobility of theA-type ions; the value of the diffusion constant iscomparable to that of liquids. The B-type ionsform a well-defined lattice and perform smalloscillations around the lattice positions. In orderto investigate microscopically such a system wehave to answer the following question: How doesa many-particle system (formed by A-type ions)with strong anharmonicities behave in a periodicpotential (formed by the B-type ious)? For ex-ample, in the case of e-AgI the A-type ions aregiven by the Ag+ particles and the B-type ionsby the I particles.

Molecular-dynamics (MD) calculations' ' areimportant in studying many-particle systems

with strong anharmonicities, such as superionicconductors, since anharmonicity is treated with-out approximations. In contrast to some analyticapproaches' ' recently employed, this methodallows one to study the particle motion moregenerally. For instance, it is possible to studyin detail the influence of the B-type lattice onthe structure and dynamics of the &-type ions, andone is able to learn something about correlationsbetween different ions; such correlations are re-flected in the frequency-dependent conductivity.In Refs. 1 and 2 we showed by MD that such effectsare very important for the description of super-ionic conductors and cannot be neglected. Untilnow it was not known how these kinds of effectscould be treated reliably in an analytical way.

In this paper we want to study collective exci-tatzons as well as the single particle motionin the Ag' subsystem of an AgI-type solid elec-trolyte. We have used the same MD model asin a previous paper' (henceforth indicated by I).Therefore this paper is a continuation of thestudy in I. It has already been shown in I thatthe MD model describes well important dynamicalfeatures of the n phase of AgI. We shall see be-low that also the behavior of the scattering func-tions (which are suitable for the description of thecollective excitations and the single particle mo-tion) are in good agreement with the correspond-ing experimental data. However, it is not our aimto reproduce in detail experimental results butto investigate some fundamental effects in con-nection with elementary excitations inherent inmany-particle systems such as superionic con-ductors.

II. MODEL

The model used in the calculation for the scat-tering functions is described extensively in Ref.

22 1058

STRUCTURE AND DYNAMICS OF SUPKRIONIC. . . . II.

1 and I. Here we only want to summarize themain characteristics of the model: N =256 ions(128 Ag' ions and 128 I ions) were arranged ina tljo-dimensional box with the same package den-sity as for the three-dimensional case (real o.-Agi)and we obtained for the length of the cell the val-ue of 62.89 A. To avoid surface effects, periodicboundary conditions were imposed on the system.For the interaction between an Ag+ ion and anyother ion (Ag' or I ) we choose the Born-Mayerpotentia1 and the Coulomb potential. The peri-odic potential is formed by a centered planar io-dine lattice. The temperature of the system was585 K; the a-phase of AgI is stable from 420 to828 K.

The reason why we had to restrict ourselvesto a two-dimensional system is discussed in de-tail in I. Furthermore, we showed in I that sucha two-dimensional AgI model is still able to de-scribe important dynamical properties of the &-phase of n-AgI. However, one should not con-clude that a two-dimensional system is able todescribe completely real n-AgI. This is becausethe structure of the two-dimensional system isdifferent from the structure of real (three-dimen-sional) a-Agl.

III. STRUCTURE FACTOR

We already studied the structure of the Ag sub-system by means of the pair correlation functionand the triplet correlation function in Ref. 7 andI. Here is a short summary of the main facts:

(i) The structure of the I subsystem is dis-tinctly reflected in the structure of the Ag' sub-system

(ii) For the Ag+ ions we have to distinguishbetween quasilattice positions and interstitialpositions. The ions situated at interstitial posi-tions are less bonded than the ions situated atthe quasilattice positions. Furthermore, theinterstitial positions are less occupied than thequasilattice positions.

(iii) For distances (Ag -Ag ) less than 2d (d isthe lattice constant of the I lattice} the systemis predominantly in a solidlike state. For distan-ces greater than 3d the system is predominantlyin a liquidlike state.

The analysis in Ref. 7 and I has been madewithin the real space. As usual we shall discussthe scattering functions within the reciprocalspace —the so-called k space, where R is the wavevector. The structure representation within thek space is given by the structure factor S(k) whichis the Fourier transform of the pair correlationfunction and can be obtained directly from the MDdata (see, for example, Ref. 8) by means of

IV. DENSITY FLUCTUATIONS

The Fourier transform of the microscopic num-ber density of a system with K, silver ions havingpositions I',.(f}, i =1, . ... ,N„is given by

(2)

To describe density fluctuations, the correlationfunction (intermediate scattering function)

F (k, f) = (p, (0)p, (f)}

is of interest. From this we obtain the coherentscattering lan& S(k, m) by

s(k)Ag -Ag

05-

1.8 2.6k(A I

FIG. 1. Structure factor for the Ag+ subsystem. Thearrows indicate the positions of the I ions.

where ( ~ ~ ~ ) denotes statistical averaging, N. is thenumber of the Ag' ions, and r,. the position vectorof the ith Ag+ ion. The experimental data "whichare relevant for the description of the scatteringfunctions of a-AgI are given as a function of theabsolute value k= ~k

~

of the wave vector. There-fore, all the results discussed below are alsocalculated for k = ~k ~; that means we have aver-aged over a sufficiently large number of vectorsk with magnitude k.

S(k) for the Ag subsystem is represented inFig. 1. It can be seen from Fig. 1 that also inthe reciprocal space the I lattice is distinctly re-Qected in the structure of the Ag' subsystem.The peak positions of our two-dimensional struc-ture factor are different from those one obtainsfor o.-AgI which is three dimensional. However,the magnitudes of S(k) obtained from the two-dimensional model are comparable to those ob-served experimentally. " This is because wechose for the two-dimensional system the samepacking density as for n-AgI.

1060 SCHOMMERS 22

CO

LL

Ll

1.0k=1.67A

' k=2.05A' k=2.4A' k=32A

0 2 4 0

2 11 05I

0.1 k=1.67A' .-

2 4 0 2 4 0 2 4t(10 sec)

1- 1--

05I 051-205A'I k=2AA' k=32A

3 0.05-

S(k, &u) =— E(k, t) exp(i ~t)dt .127r

S(k, &u) is proportional to the coherent differentialscattering cross section" and can be obtained fromneutron scattering experiments.

The MD results for E(k, t) are represented in

Fig. 2 for several wave numbers k. It can beseen that for all k the intermediate scatteringfunction exhibits two distinct contributions: anarrow line on top of a much broader centralline. This behavior is also reflected in the co-herent scattering law S(k, &u). However, becauseof the Fourier transformation the narrow peakin F(k, t) is predominantly reflected in the broaddistribution of S(k, &u) and the broader central lineof F(k, t) is predominantly reflected in the narrowpeak of S(k, &u). The broad distribution of S(k, ~)is relatively weak at low k and becomes moreand more pronounced as k increases.

It is interesting to note that spectra of the formgiven in Fig. 2 have also been observed by inelas-tic neutron scattering experiments" on cx-AgI.Also these experimental results exhibit the twocharacteristics we have drawn from our MD data:(i) the spectra contain a narrow and a broad com-ponent, and (ii) the broad distribution becomesmore and more pronounced with increasing wavenumber k. The MD results given in Fig. 2 can-not be compared directly with the correspondingexperimental data because the two-dimensionaland the three-dimensional k space are not com-patible with each other. However, we shall seein Sec. VI that the calculated and the experi-mentally observed frequency spectrum f (&)agree well; f (~) can be extracted from experi-mental data by averaging of the coherent scat-tering law S(k, v) over a sufficiently large re-gion of k space.

How can we interpret the behavior of E(k, t)and S(k, ~)? We pointed out in I that within theMD model used here well-defined jump-diffusionprocesses take place, and we want to investigatehow the features of E(k, t) and S(k, w) are deter-mined by the characteristics of the jump-dif-fusion process. In order to decide on whattime scale the translational diffusion takes place,we have calculated F(k, t) and S(k, ~) by meansof the simple diffusion model. For this, one ob-tains'

and

F(k, t) = exp(-Dtk') (5)

1 Dk2

7r (Dk) +&8' ' (6)

where D is the diffusion constant, which describesthe translational random walk from one restrictedregion (where the "local" motion takes place) toanother. We calculated D from the mean squaredisplacements (see I) and obtained D = 2.85 && 10 'cm'/sec; this value is close to that observedexperimentally" (-2.6 &&10' cm'/sec). The re-sults for the diffusion model are represented inFig. 2 and by comparison with the MD data wemay conclude that for all wave numbers k thetime behavior of the narrow line in E(k, t) isdetermined by the translational diffusion process.This means, that the broad peak in S(k, &u) shouldmainly reflect the translational diffusive motion;the "local" diffusive motion within restricted re-gions of space should be reflected in the broadline of E(k, t) and in the narrow peak of S(k, ~),respectively. More details concerning the jump-diffusion process are discussed in I.

We have also performed an MD calculationzeithout the presence of the Coulomb interaction.In this case we have a many-particle system con-sisting of gveak disks in a periodic potential (seealso the discussion in I). From the analysis ofthe microscopic square displacements we werenot able to recognize jurnp-diffusion processes inthe motions of the weak disks. Furthermore, thediffusion model does not fit any part of the inter-mediate scattering function E(k, t) (see Fig. 8).Therefore, we may conclude that the presence ofthe Coulomb interaction is of considerable im-portance in the description of the typical fea-tures of superionic conductors.

0 2 4 0 2 4 0 2 4 0 2 4

+ (10"sec' j

FIG. 2. Intermediate scattering function E(k, t) andscattering law S(k, ~) for the Ag' subsystem. Solidlines, MD results; dotted lines, diffusion model.

V. INCOHERENT INTERMEDIATE SCATTERINGFUNCTION

Whereas the coherent intermediate scatteringfunction F(k, t) describes the collective behaviorof the system, the incoherent intermediate scat-

22 STRUCTURE AND DYNAMICS OF SUPERIONIC. . . . II. 1061

1.0C)

—0.5

0 6 12 0 6 12t(10' sec)

1.0

k='l.2 A"

x x x x xix

~ ~ ~ ~ e ~~ ~

C3

LL—05

A"

k =1.67A

0 6 12 0 6 12

t(10 Gs&c)'

C0~ 0.5-C

x x x x x xxx x

~ ~ ~~ ~ ~

FIG. 3. Intermediate scattering function F(k, t) for a.

many-particle system consisting of weak disks. Solidlines, MD results; dotted lines, diffusion model.

tering function"

(k,i) i+exp=(i—k [r,. (t) —F, ( )]0})

reflects the individual motion of an Ag' ion.Therefore, by comparison of E,(k, t) with E(k, t)we are able to learn something about the couplingbetween the individual motion and the collectivemodes of the system. The MD results for E,(k, t)are represented in Fig. 4 for several values ofk as a function of time. The values k =1.2 and2.4 A ' span the whole region of structural detailin the structure factor S(k) (see Fig. 1). FromFig. 4 we can see the following:

(i) The time behavior of E,(k, t) and E(k, t) isvery much the same; therefore, we may concludethat the individual motion is strongly coupled tothe collective modes of the system (see also thediscussion in I). Because of the long-range na-ture of the Coulomb interaction we may assumethat the single-particle motion is due to the col-lective modes of the system; this assumption issupported by the analysis given in Sec. VI.

(ii) The values of E,(k, t) and E(k, t) approacheach other with increasing wave number k; itis well known" that in the limit of 0 -~, inter-ference effects disappear and E(k, t) -E,(k, t).

(iii) The coupling between E,(k, t) and E(k, t) isinfluenced by the structure factor S(k). This be-havior can be described schematically as follows(see also Fig. 1):

S(k) & 1-E(k, t) & E,(k, t),S(k) =1-E(k, t) =E,(k, t),S(k) &1-E(k,t) & E,(k, t) .

Because of this behavior we have calculated E,(k, t)by means of the well-known convolution approxi-

CA

1.0 iOI(D

— 0.5-(UE

1.0'

k=2.05 A"

~ ~x x

k=2.4 A"

0.5- i

X ~

2 3t{10' sec}

5

m ation":

E,(k, t) =E(k, t)/S(k) .It can be seen that already this simple ansatz isable to describe quite well the coupling betweenthe collective and single-particle modes of thesystem.

VI. OPTICAL MODES AND SINGLE-PARTICLEMOTION

The single-particle motion can also be studiedby the frequency spectrum of the velocity auto-correlation function and is given by

2 ""f (u) =— 0'(t)cosset dt,

7T Qp

where

4 (t) = (U, (0) ~ U.(t) )/( U.(0)') .

FIG. 4. Solid line, F,(k, t); crosses, E(k, t); Ml points,points, convolution approximation.

1062 W. SCHOMMKRS

U,(t) is the velocity at time t for one Ag' ion of theensemble. f (v) is normalized to unity.

The frequency spectrum for n-AgI has beenextracted from the experimentally observed scat-tering law S(k, ur) by Suck" using Bredov's methodof averaging. " As can be seen from Fig. 5(b) theagreement between the experimental data and thecalculated values is very good.

It is interesting to note that Funke et al.' ob-served in P-AgI optical phonons with frequencieswhich are very close to the position of the mainpeak of the frequency spectrum [see Fig. 5(a)].This suggests, together with the discussion inSec. IV, that in o.-AgI the oscillatory componentin the single-particle motion is mainly describedby the optical phonons if we assume that the forceswhich are responsible for these kinds of modesare approximately the same in both phases ofsilver iodide; Buhrer et al."already pointed outthat the phonon spectrum of P-AgI is also of im-portance for ionic dynamics of o.-AgI. In the fol-lowing section we shall see more directly that theoptical modes in P-AgI observed by Funke et al.'are also typical modes in u-AgI.

phonons or phononlike excitations because theyare covered by the diffusion modes (these arenonpropagating modes) and these are due to thestrong anharmonic behavior of the Ag subsystem.In order to answer the question whether the opticalmodes observed in P-AgI by Funke et aE. are alsotypical modes in n-AgI we have to select thesemodes from the other excitations. The followingway is suggested.

Optical modes are characterized by the collec-tive oscillation of both species (Ag' and I ) againsteach other. In this case the total velocity of theAg' subsystem

I.(t) = g U;.(t) (10)i~

oscillates against the total velocity of the I sub-system

T(t)= QU-, (t).

Because of the conservation of total momentum(there are no external forces acting on the sys-tem) we obtain in the center-of-mass system

VII. CONSTRUCTION OF ELEMENTARYEXCITATION S

1,(t) =- -1(t). (12)

E(k, t) and S(k, &u} only exhibit a small struc-ture (see Fig. 2); we are hardly able to recognize

Let us now define a microscopic number densitywhich is correlated to I,(t) in the following man-ner:

+ j=l +

That means the wave vector k is projected in thedirection of I,(t =0). The corresponding inter-mediate scattering function is given by [see Eq.(3)]

E„(k,t) = (p"~(0) p~(t)}

and the scattering law is

(14)

S„(k,&u) =—~ E„(k,t) exp(i~t)dt .w CO

(15)

0 2 4 6 8 ')0 12

+()0"sec ')FIG. 5. (a) Dashed line, dispersion curve for optical

phonons in P-AgI {from experiment); solid line, MD re-sults for the dispersion curve for optical elementaryexcitation. s in n-AgI constructed in Sec. VII. (b) Solidline, MD results for the frequency spectrum f(~);crosses, experimental results; full point, obtainedfrom the experimentally observed diffusion constant.

The MD data show that I,(t) does not prefer anydirection in space; the vectors I,(t) at differenttimes form an isotropic distribution in space.Therefore, E„(k,t) and S„(k,to) only depend on themagnitude of the wave vector k.

The MD results (examples are given in Fig. 6)show that E„(k,t)—in contrast to E(k, t)—stronglyoscillates, and that the scattering law S„(k,&u}

exhibits relatively sharp peaks. The dispersioncurve of the main peak of S„(k,e) is representedin Fig. 5(a); it can be seen that this dispersioncurve is very close to that observed for opticalmodes in P-Agl (see Ref. 9). Therefore, we mayconclude the following:

STRUCTURE AND DYNAMICS OF SUPERIONIC. . . . II. 1063

C)

1.0i

0.5

k-23A-'

0 2 4 0 2 4t{10 sec)

k=1.67A k=2.3A

(i) By means of the collective coordinate de-fined by Eq. (13) we are able to select phononlikeexcitations geithout using a reference lattice.

(ii) The forces which are responsible for theoptical modes in P-AgI are approximately thesame in n-'Agl (see also Ref. 1V).

Besides the main peak, other peaks with lowerintensity appear in S„(k,~). This is because thedisorder in n-AgI breaks the symmetry of theunit cell, leading to additional modes.

The actual direction of 1,(t) [see Eq. (10)) is ingeneral unknown and, therefore, it is not pos-sible to measure S„(k,&u); only in the case of aone-dimensional system is the direction of I.(t)known. However, as already discussed by Rah-man, "MD models are capable of giving thecow.p/ete structural and dynamical informationof the system under investigation and it is there-fore possible to investigate any useful correla-tion function, which, till now, has not appearedin the formal structure of the statistical mech-anics of condensed matter. In other words, com-puter "experimental" investigations make it pos-sible to extend the list of useful correlation func-tions. E„(k,f) is an example.

0 2 4 6 0 2 4 6~(10 sec j

PEG. 6. Intermediate scattering function E~~

(k,t) andscattering law S~~ (k, (d).

VIII. SUMMARY

In this paper we have studied important dynami-cal properties for superionic conductors. Inparticular, we have investigated the behavior ofa disordered many-particle system (Ag') withstrong anharmonicities in a periodical potential(I ). We used a two-dimensional model (con-sisting of 256 particles) because both the well-defined I lattice and the Coulomb potential giverise to long-ranged correlations in the disorderedAg' subsystem and it turned out that the size ofa three-dimensional model consisting of the samenumber of particles (256) is too small in orderto describe the correlations well.

In summary, the main results of the calcula-tions are as follows:

(i) The coherent intermediate scattering func-tion E(k, f) exhibits two distinct contributions:a narrow line on top of a much broader line. Thisbehavior is also reflected in the scattering law

S(k, ~) and is supported by recent experimentalresults.

(ii) The shape of the spectra can be interpretedin terms of jump-diffusion processes.

(iii) The time behavior of the incoherent inter-mediate scattering function E,(k, f) is very similarto that observed for E(k, t); this means that theindividual motion is strongly coupled to the col-lective modes of the system. The coupling be-tween E, (k, f) and F(k, t) is dependent on the struc-ture factor S(k) and it turned out that already theconvolution approximation describes this coup-ling well.

(iv) By means of the frequency spectrum ofthe velocity autocorrelation function we wereable to conclude that at least part of the single-particle motion is due to optical modes.

(v) We have constructed optical elementaryexcitations geithout using a reference lattice;the dispersion curve of these excitations is veryclose to that for optical phonons observed in P-Agi.

Although we had to restrict ourselves to a two-dimensional model the results for the main dy-namical properties of the system agree well withvarious experimental data for n-AgI.

ACKNOWLEDGMENT

The author would like to thank M. Kobbelt forhelpful discussions and his critical reading of themanuscript.

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