Infrared Phys. Vol. 32, pp. 425433, 1991 0020-0891/91 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright Q 199 I Pergamon Press plc

A GENERALIZED LYDDANE-SACHS-TELLER RELATION FOR SOLIDS AND LIQUIDS

A. J. SIEVERS’ and J. B. PAGE’ ‘Laboratory of Atomic and Solid State Physics and Materials Science Center, Cornell University, Ithaca, NY 14853-2501 and rDepartment of Physics, Arizona State University, Tempe, AZ 85287-1504, U.S.A.

Abstract-A review is given of a recently developed generalization of the Lydanne-Sachs-Teller (LST) relation, which provides a useful connection between the long-wavelength dynamical behavior of nonmetallic condensed media and their static and high-frequency dielectric properties. The characteristic dynamical frequencies are defined in terms of second moments of the relevant response functions, and very general causality arguments are used to obtain the generalized LST relation, which is applicable to composite and disordered media. Results are discussed for both the bulk medium case and for small particles. For crystalline solids, one recovers the original LST relation, in terms of long wavelength optic mode frequencies. The generalized LST relation is illustrated via computer simulations of a hypothetical composite dielectric which has one component near a ferroelectric phase transition, using the Effective Medium and Maxwell-Garnett approximations for the dielectric response. It is also illustrated and discussed in terms of recent laboratory data on ZnSjdiamond composite media. Finally, the generalized LST relation is used to obtain a new generalization of the Clausius/Mossotti relation.

1. INTRODUCTION

The Lyddane-Sachs-Teller (LST) relation”) played an important role in the early interpretation of displacive ferroelectricity. Frdhlich(*) and Cochran”) first recognized that an IR active optic mode must be temperature dependent near the transition temperature. For a cubic crystal with a single IR active vibrational mode the appropriate long wavelength dielectric function is

Evaluating this relation at zero frequency gives the LST relation

which provides a simple connection between the long wavelength transverse and longitudinal optic mode frequencies, o, and o,, and the dc and optical dielectric constants, 60 and cG: in a diatomic insulating crystalline solid. [In the small damping limit where y G w,, o, and o, give the pole and the zero of c(o).] Experimentally, the temperature dependence of the dc dielectric constant in the paraelectric phase near T, of a ferroelectric crystal is observed to follow a Curie-Weiss law

According to Ref. (3) this relation together with equation (2) suggests that 0: = A(T - T,), and soft mode behavior must occur. Equation (2) has been generalized to cubic crystals with more than two atoms per unit ce1P3) and also to include damping and anharmonicity.(4-6’ In addition Barker’5,6’ has found that equation (2) can be obtained from a causality argument if the response is approximated by a d-function mode at 0,. All of these extensions treat single crystals.

At the same time that progress was being made to better characterize the LST relation, more precise experimental measurements on the modes of displacive ferroelectric crystals have tended not to agree with the LST predictions. (‘N A number of experiments now indicate that near T,, temperature-dependent changes in the dc dielectric constant are observed, although the soft mode frequency is found to be temperature independent over the same interval. (*) The classic ferroelectric BaTiO, is a case in point since the soft mode frequency remains unchanged at about 60 cm-’ over a 100 K temperature interval near the cubic-to-tetragonal phase transition, even though the dc

INF 32,BI--BB 425

426 A. J. SIEVERS and J. B. PAGE

dielectric constant continues to increase as T, is approached. c9) The conclusion is that near the transition temperature an additional relaxation mechanism is dominant and provides most of the temperature dependence of the dc dielectric constant, with the end result being similar to that found in hydrogen bonded ferroelectric crystals. (‘) Such additional relaxation phenomena are not included in the LST soft mode description. It now appears that the LST relation does not provide the necessary connection between the static and dynamic properties of actual displacive ferroelectrics. However, one may ask if a more general LST relation exists which could provide an exact connection between the static and dynamic properties of these more complex systems, thus providing a clue as to the important temperature-dependent dynamical feature to monitor near the phase transition temperature. Clearly, if relaxation dynamics are the controlling mechanism, it makes little sense to focus exclusively on the temperature dependence of optic mode frequencies near T,.

Recently, it has been found that such a generalized LST relation does exist,(‘0-‘2’ and this development is reviewed here. The derivation is sufficiently general to apply to all linear non-conducting dielectric systems, including ferroelectrics, liquids and inhomogeneous media. The key to the generalized LST relation is in the identification of the characteristic frequencies themselves. One way to identify the characteristic frequencies has been described in Ref. (5), where a number of modes are defined, each represented by a complex frequency. However, this method does not generate a universal LST relation. Another more general way to identify the characteristic frequencies of an arbitrary system is with a moment representation,(‘3) and it is shown here that appropriately defined moments do provide the correct framework for uncovering the generalized LST relation.

2. MOMENT REPRESENTATION OF CHARACTERISTIC FREQUENCIES OF BULK MEDIA

Before one can make contact with the moment representation, it is important to define the appropriate macroscopic response functions of the general system. For a bulk nonmagnetic linear isotropic dielectric, let the external susceptibility be defined as P = xeXtEeX’. Only two kinds of response functions can occur since, by Helmholtz’s theorem, there are only two kinds of vector polarization fields, transverse (solenoidal) and longitudinal (irrotational), in the system. It can be shown readily that the transverse response, in terms of the macroscopic dielectric function t(e)), istill

x:“‘(w) = [E(W) - 1]/47c (4)

and the longitudinal response is

x;“‘(w) = [1 - l/C(O)]/47c. (5)

It follows from an argument like that preceding equation (3) of Ref. (12) that the loss functions for the transverse and longitudinal responses are (47~)’ Im[c(o)/t,] and (47~))’ Im[-c, it(w)], respectively.

With the LST relation in mind, the two characteristic frequencies are defined in terms of weighted second moments of the longitudinal and transverse loss by””

and

(6)

(7)

In terms of these two quantities it has been determined that the generalized LST relation is””

(8)

Lyddane-Sachs-Teller relation for solids and liquids 427

Let us see how this expression follows directly from the Kramers-Kronig relations. Recall the KK relation

(9)

Assuming that the loss falls off sufficiently fast at high frequencies, we have the high frequency approximation for this KK relation

Re 1 1 t(X) _ 1 - em

x --$i j:dww Tm[F]. (10)

Using this result to evaluate equation (5) for x;“’ in the high frequency limit, we obtain

Next we note that since xf”‘(~) is a causal response function, it also satisfies the KK relation so that

Re[x;“‘(x)]-xyi=l o s

a, dew 7 LmW’(x)l; w--x

hence

Re[Xt”‘(x)] - x;“A = -&- f cc s 0a s Im 3 . [ 1 E(O) For high frequencies, this equation gives

(12)

(13)

(14)

where it has again been assumed that the high frequency loss falls off sufficiently fast. Comparing equations (11) and (14) one obtains

(15)

Multiplying this equation through by 2/n gives familiar expressions: they are the integrals associated with they-sum rules for the appropriate response functions. (14) This equality in equation (15) shows that the numerator of equation (6) is equal to the numerator of equation (7) hence the ratio of these last two equations reduces to

&?m($)

(02” j: $ fm($$) ’ (16)

Next, equation (9) is evaluated at w = 0 to obtain the polarizability sum rule for the transverse response, namely

A!?-l=~J~!!.f*m[L~]~ cm

Similarly, the corresponding polarizability sum rule for the longitudinal response is

(17)

428 A. J. SIEVERS and J. B. PAGE

But equations (17) and (18) are just the numerator and denominator of equation (16). With these substitutions this equation reduces to

the desired LST-like relation.

(19)

3. COMPOSITE MEDIA TESTS OF THE GENERALIZED LST RELATION

3.1. Computer simulation

The long wavelength responses of inhomogeneous dielectric media are ideal for testing the generalized LST relation. Depending on the topology of the structure, two different mean-field calculational methods are available to describe both the dc and optical response of composites, the Maxwell-Garnett (MG) and the Bruggeman (EMA) models.“”

In the simulation the vibrational response of a two component composite is considered, one component being inert (6, = c,, = 10) and the other obeying equation (1) (parameter values: w, = 5 cm-‘, w,/r = 1, o,/o, = 100 and 6 bco = c,, /2 = 5). These values, which represent the extremes of the lattice dynamics possibilities, would be consistent with a particle mixture of, say, Si (the inert material) of volume fill fraction .f and a displacive ferroelectric, at a fixed temperature just above the phase transition value.(‘O’

For both models we calculate to/t, and also evaluate equations (6) and (7) as a function of$ Both moments are finite. The root mean square moment frequencies for the EMA are presented in Fig. l(a). Note that ((o~),)‘~~ is fairly constant in frequency with increasingfuntilf v 0.6, after which it increases rapidly. This change has to do with the disappearance of the pole at o, in the composite forf’ > 2/3. (“) On the other hand, ((02),)“’ shows a much weaker dependence onfover the entire range. Both frequencies converge to a single value at largef, namely to the isolated sphere resonance frequency of the nearly ferroelectric material embedded in the Si medium.

In Fig. l(b) the solid curve gives the calculated dependence of Lo/t,, onf, and the solid points give the dependence of (02),/(o’),, for the EMA model. The corresponding calculations for the MG model are also shown in this figure. Because the two components are treated in an asymmetric way in the MG model, there are two possible arrangements, one for medium a surrounded by b and vice versa. Both possibilities are presented in Fig. l(b). The dashed and dot-dashed curves correspond to co/t,, and the solid squares and triangles correspond to the values of (o’),/(02),. Although the EMA and MG models distribute the spectral weight of the transverse and longitudinal response functions in different ways, the second moment expression, equation (8) holds independent of these topological details. This reflects the general nature of equation (8) which we have obtained here from very general arguments applied to causal response functions.

3.2. Experimental far IR study

Far IR measurements on ZnS:diamond composites have provided the first experimental demonstration that the original LST relation is too specific to describe disordered systems.“‘.“’ Since only the ZnS component produces IR activity, with a reststrahlen band centered near 300 cm-‘, the ZnS:diamond composites provide a simple and nearly ideal extension away from single crystals to a more complex disordered system for testing the generalized LST relation. By means of far IR reflectivity measurements and a Kramers-Kronig analysis on samples with a volume fill fraction of up to 55% diamond, the required transverse loss function, Im(c), and the longitudinal loss function, Im( - l/c), have been obtained with which to calculate the characteristic second moment frequencies of the medium.

These experimental results are shown in Fig. 2(a) and (b). In both cases, as the diamond concentration increases, the strengths of the narrow loss peaks, located near 275 and 350 cm-‘, respectively, decrease in height and broaden; however, inspection of the two frames shows that the peak positions themselves, co, and o,, remain relatively unchanged. Each loss function broadens

Lyddane-Sachs-Teller relation for solids and liquids 429

104 A N

3’ - >=

103

(v I

3 I

- 102

6 (38 10’

\ w”

loo_ 0.0 0.2 0.4 0.6 0.8 1.0

f Fig. 1. Some calculated second moment and dielectric constant properties of a hypothetical composite medium according to the EMA and MG models. (a) Dependence of the root-mean-square frequencies on f for the EMA model. Triangles: ((w’),)‘/~. Solid points: ((w*),)‘/‘. (b) Dependence of the ratio of the second moments and of the dc dielectric constant onf: EMA calculations: solid curve, ~,,/t,, solid points, (02),/(02), MG calculations: inert material embedded in the near-ferroelectric: dashed curve, co/c,, solid trangles, (wz),/(w2),. The dot-dash and squares are similar calculations but for the near-ferroelec-

tric embedded in the inert material. (After Ref. IO.)

r

200 300 400 Frequency (cm-l)

Fig. 2. Frequency dependence of the imaginary part of the response functions obtained from a Kramers-Kronig analysis of the reflectivity data for ZnS:diamond composites. (a) Transverse field case. (b) Longitudinal field case. The volume fill fraction of diamond ranges from 0 to 55% in steps of 11%

with the solid line f = 0 and the dash-dash-dot line f = 55%. (After Ref. 18.)

430 A. J. SIEVERS and J. B. PAGE

in an asymmetric way, with the expanding wing extending toward the frequency peak of the other type of loss function.

Figure 3(a) shows the measured values of o:/w: (triangles) and co/cm (squares) at various fill fractions. The values of co/c,, which are determined by fitting the general shape of measured reflectivity using a single Lorentz oscillator model, show a systematic decease asf increases, while the values of of/wf are nearly independent off. Clearly, the LST relation connecting ~~/t, and w:/o: is not valid for composite materials. The EMA model also produces the same discrepancy between OF/O: (dashed line) and co/c= (solid line) in these ZnS:diamond composites.

With the second moment integrals [equations (6) and (7)], (o*), and (w*), are then calculated, the ratios (02),/(w2), are found (circles), and the results are compared with the measured values of tO/cX (squares) at various fill fractions in Fig. 3(b). The excellent agreement between co/c, and (w*),/(o*)~ demonstrates that within the experimental uncertainties the generalized LST relation holds for these ZnS:diamond composites.(i*’

4. CHARACTERISTIC FREQUENCIES OF A DISORDERED SOLID OF RESTRICTED SIZE

A linear nonmagnetic macroscopic ellipsoidal dielectric particle embedded in an inert dielectric ch is considered. A uniform applied electric field ETpp is applied along one of the principal axes of the ellipsoid characterized by depolarization factor Ni, where N, + N, + NJ = 471. For the limit where the wavelength is much larger than the optical dimension of the particle, the polarization density of the particle in the ith direction is(“)

p, = xV’&-‘P

where

9 - 0.0 0.2 0.4 0.6 0.6 1.0 -

f Fig. 3. Fill fraction dependence of to/e, and the ratio of the squares of the characteristic loss function frequencies of 2nS:diamond composites. Thefvalues are the same as in Fig. 2. (a) Comparison between measured eo/tm (squares) and of/w: (triangles), where w, and mt are the respective frequencies at which the longitudinal and transverse loss functions are a maximum. The EMA predictions for sa/c, (solid line) and c$/c$ (dashed line) are in good agreement with experiments. (b) Comparison between the measured co/e= (squares) and (w2),/(w2), (circles), where (w2), and (o*), are the characteristic longitudinal and transverse squared frequencies based on the second moment representation, The agreement between these quantities validates the generalized LST relation for inhomogeneous 2nS:diamond composites. The EMA prediction is shown as a solid line. Note: although for visual clarity the error bars for co/s, are not shown,

they are the same as in (a). (After Ref. 18.)

Lyddane-Sachs-Teller relation for solids and liquids 431

Our Eapp is the field in the embedding medium in the absence of the particle. Thus for the c(o)+c,, limit of no particle in the embedding medium, ~$~P(w)-+(c~ - 1)/47r = xh, where xi, is the ordinary susceptibility of the embedding medium, relating the polarization to the total field.

Equation (20) gives

Im[xfPP(W)] = - 4~~1m[~(W)+&- *)I.

(21)

Because c,, is real, we have Im c(w) = Im[c(W) + ch(47r/Ni - I)]. Accordingly, we define q,(o) = t(m) + ch(47r/Ni - 1). Since equation (21) is proportional to the small particle loss, the corresponding second moment expression can be written as(r2)

In this notation equation (7) becomes

(22)

(23)

Comparison with equations (6) and (7) shows that (o’)~, and (o*), above are formally identical with, respectively, the longitudinal and transverse second moments (o*), and (o’), for a bulk medium of dielectric constant vi(o). Applying equation (8) to vi(o), we immediately have

(r,,2\ M. /da \ ’ \- /I ‘,Icc

% +ch(+ ‘)

For a sphere, N, = N? = NJ = 4n/3 and

L,, + 26, =---. t, + 26,

(24)

(25)

Note that the right hand side of this general LST-like expression is the same as that obtained from a small single crystal sphere which has a single IR active mode.(‘)

5. DISCUSSION AND CONCLUSIONS

The far IR measurements on ZnSdiamond composites show that the generalized LST relation does correctly make the connection between the static and electrodynamic properties in disordered and inhomogeneous materials. Since only the ZnS component produces IR activity, this composite system provides a simple and nearly ideal extension away from single crystals to a more complex disordered system for a quantitative test of the generalized LST relation. The required loss functions with which to calculate the characteristic frequencies of the medium are obtained from far IR reflectivity measurements and a Garners-Kronig analysis. The results show that the simple connection between the static and high frequency dielectric constants and dynamic frequencies can be made only when these frequencies are obtained using the second moment integrals given by equations (6) and (7).

In the spirit of the original LST analysis of the temperature dependent optic mode dynamics of bulk displacive ferroelectric crystals, one can look for the vibrational implications of the generalized LST relation equation (19) but now the constraints are more subtle. A strong temperature dependence in co translates into an equally strong dependence in (o*), but what exactly does this mean? The definition of an rms width, At = (<w*), - (o):)“’ provides a standard use of a second moment. Note that if Im[c(w)] displays a simple delta function-like response at the frequency ol, then (o*), = (0): = o:, A:-+0 and the original LST relation, equation (2), is recovered; however, when this approximation is not valid, i.e. for a mixed Lorentz and Debye-like(“) or more complex

432 A. J. SEVERS and J. B. PAGE

response, a temperature dependence in c,, could signify a temperature dependence of the mean frequency of the general transverse response (o), and/or the rms width A,.

One can use the second moment representation outlined here to make contact with the single crystal Clausius-Mossotti relation. (*Q The difference between the longitudinal and transverse second moment expressions defined by equations (6) and (7) is

where 0; is the macroscopic plasma frequency. Inserting equation (8) into this expression gives

(27)

for the bulk material. For a small sphere of dielectric constant c(w) embedded in a host dielectric c,,, we can eliminate (o*), in equation (25) by means of equation (27) to give

60 - G 1 -----= 60 + 2Eh ( Y t, + 26, <w”i’,,, ’ (28)

Now if the response is measured over a large enough frequency interval so that cm-+ 1 and the sphere resonance is calculated for a particle in vacuum (c,, = 1), equation (28) becomes

co-1 1 o2 -c-2 c0 + 2 3 <w2)sp

(29)

which is a generalized Clausius-Mossotti relation. For a cubic crystal, microscopic theories’*‘) can be used to show that o~/(o~),,+~~~((oI/u) where a is the polarizability and v the volume of the unit cell, but for the general case considered here such a detailed connection between macroscopic and microscopic is not yet possible. Still, for lack of a better alternative it is reasonable to define a microscopic “effective polarizability per unit cell” for a disordered system by first finding t(m) from experiment or theory and then calculating ok/,, with equations (26) and (22).(**)

Although the discussion here has tended to emphasize only the solid state lattice dynamics context since that is where the LST relation was discovered, the results equally well apply to non-conducting liquids and also connect the optical and static properties of other degrees of freedom, such as the electronic transitions. The general conclusion is that in the small signal limit (within the bounds of long wavelength linear response theory), a generalized LST relation must exist for an arbitrary condensed matter system which may include anharmonic interactions, be ordered or disordered, or be homogeneous or inhomogenous.

Acknowledgernenfs-One of us (A. J. S.) would like to acknowledge helpful discussions with T. W. Noh. This work was supported by NSF-DMR-88-18558 and AR0 # DAAL03-90-G-0040.

I. 2. 3 4: 5. 6. I. 8. 9.

10. II. 12. 13.

14.

REFERENCES R. H. Lyddane, R. G. Sachs and E. Teller, Phys. Rec. 59, 673 (1941). H. Frohlich, Theory qf Dielecfrics, Clarendon Press, Oxford (1949). W. Cochran, Phvs. Rea. Lerf. 3, 521 (1959); W. Cochran, Adll. Phvs. 9, 387 (1960). A. S. Barker, Jr, Ph~s. Reo. A 136, 1290 (1964). A. S. Barker, Jr, in Ferroelecfrics (Edited by E. F. Weller), p. 213. Elsevier, Amsterdam (1967). A. S. Barker, Jr, Phys. Reti. B 12, 4071 (1975). G. Burns and B. A. Scott, Solid St. Commun. 13, 417 (1973). 3. P. Sokoloff, L. I. Chase and D. Rytz, Phys. Reo. B 38, 597 (1988) and references therein. Y. Luspin, J. L. Servoin and F. Gervais, J. Phys. C 13, 3761 (1980). T. W. Noh and A. J. Sievers, Phys. Rev. Lert. 63, 1800 (1989). A. J. Sievers and J. B. Page, Phys. Rer. B 41, 3455 (1990). A. J. Sievers and J. B. Page, Phys. Rev. B 41, 12 562 (1990). With a different moment definition M. F. Thorpe and S. W. deLeeuw have produced the LST relation for the special case of a classical harmonic disordered solid [equation (74) in Phys. Rev. B 33,849O (1986)]. It is important to recognize that the moments defined by their equation (64) with n > 0 are infinite for a general system with anharmonicity. Theoretically, the infinity can be subtracted out by calculating the difference between longitudinal and transverse moments as the authors have done, but since experimental data are only known over a finite frequency range they cannot be analyzed this way. M. Altatelli, D. L. Dexter, H. M. Nussenzveig and D. Y. Smith. Phps. Ret B 6, 4502 (1972).

Lyddane-Sachs-Teller relation for solids and liquids 433

15. R. Landauer, in Proceedings of the First Conference on the Electrical and Optical Properties of Inhomogeneous Media (edited by J. C. Garland and D. B. Tanner), AIP Conf. Proc. No. 40 (American Institute of Physics, New York, 1978). p. 2.

16. D. Stroud, Phys. Rev. B 19, 1783 (1979). 17. T. W. Noh, A. J. Sievers, L. A. Xue and R. Raj, Optic Lett. 14, 1260 (1989). 18. S. A. FitzGerald, T. W. Noh, A. J. Sievers, L. A. Xue and Y. Tzou, Phys. Rev. B 42, 5469 (1990). 19. For Debye loss both sides of equation (15) are infinite, a nonphysical result reflecting the invalidity of the Debye

dielectric response function at high frequencies. Numerous experiments have shown that the Debye dielectric function is a poor approximation for describing the high frequency response of real systems. See B. K. P. Scaife, Principles OJ Dielectrics, p. 72, Clarendon Press, Oxford (1989); G. W. Chantry, Submillimeter Spectroscopy, p. 181, Academic Press, New York (1971).

20. M. Born and K. Huang, Dynamical Theory of Crystal Lattices. Oxford Press (1956). 21. N. W. Ashcroft and N. D. Mermin, Solid St. Phys. p. 542. Sanders College, Philadelphia, Pa (1976). 22. If the electronic degrees of freedom are well separated in frequency from the vibrational ones and the electronic

polarizability per unit volume of a disordered system is of interest than co in equation (29) is the “static” contribution from the electronic terms; i.e., it is equal to the squared index of refraction evaluated at an intermediate frequency between the vibrational and electronic transition frequencies. The lower limit for the frequency integrals in the moment expressions should begin here.