Solid State Communications, Vol. 31, pp. 663-666. Pergamon Press Ltd. 1979. Printed in Great Britain.

DIELECTRIC BEHAVIOUR OF ALKALINE EARTH OXIDES AND SULPHIDES

Jai Shanker and V.C. Jain*

Department of Physics, Agra College, Agra-282002, India

(Received 22 February 1979 by A.R. Verma)

Dielectric behaviour of MgO, CaO, SrO, BaO, SrS and BaS has been studied on the basis of the shell model and the exchange charge shell model. Values of polarizabilities and effective charge parameter calcu- lated from the two models are compared with experimental data.

THE PHYSICS of alkali halides and the tetrahedral semiconductors has been the subject of numerous investigations in last two decades, but little is known about the alkaline earth chalcogenides (AEC). These materials are very interesting from a theoretical point of view because they provide a link between the highly ionic alkali halides and the largely covalent I I I - V com- pounds. These substances are also of technological interest because they are highly refractory materials. In particular the oxide crystals doped with transition ele- ments may be of some use m laser development [1]. From this point of view the studies on the dielectric behaviour of these crystals become of considerable importance.

In the present paper we adopt a both ion polariz- able shell model since the one ion polarizable shell model usually adopted in lattice dynamical calcula- tions [2-4] is not adequate for the crystals under study (except for MgO). This is because the polanz- abilities of cations in AEC are not negligible as com- pared to those of anions. It should be mentioned that recently Sangster [5] has demonstrated the consistency of the shell model with recent experimental data for alkali halides. Measurements of dielectric data for AEC have been very scanty. We have been able to collect the data for six AEC crystals, viz. MgO, CaO, SrO, BaO, SrS and BaS. An analysis of the dielectric properties of these crystals is presented here within the framework of the shell model ana the exchange charge shell model.

In the shell model the electronic polarizability of an ion in free state is expressed as

y2 e2 a r - k (1)

where Ye is the shell charge and k is the spring constant coupling the shell and core. When ions are considered in

* Permanent address: P.D. College, Shikohabad (U.P.), India.

the crystal, the equations of motion of their shells and cores lead to the following relations [6]

(y++z)2e 2 (y_--z)2e 2 z2e 2 a o - + + - - ( 2 )

k+ k_ f

(y+e) 2 (y_e) 2 kk+ - - - ( 3 )

e*/e = 1 + (4)

Here ao, a** and e*/e are, respectively, the tow fre- quency polarizability, high frequency polarizability and effective charge parameter.y_+ are the shell charges and k_+ are the spring constants of cations (+) and anions (--). Ze is the ionic charge, f i s the short range force constant which is related to the compressibility 3 as follows

6to f = - - (5)

where ro is the equilibrium nearest neighbour separa- tion. Equation (5) is valid for NaC1 structure only. Values of the shell model parameters for alkali and halogen ions have been given by Dick [7]. It has been suggested by Dick and Overhauser [8] that the shell charges ye for different lsoelectronic ions are same. Thus for alkaline earth or chalcogenldes ions we can take ye as that for the corresponding isoelectronic alkali or halogen ions. Values of the spring constants are then determined from the free ion polarizabilities [9] using equation (1). The shell model parameters so obtained are given in Table 1. We have calculated ao, a~, and e*/e from equations (2)-(4). The results are compared with experimental data in Table 3. Values o¢

663

664 DIELECTRIC BEHAVIOUR OF ALKALINE EARTH OXIDES AND SULPHIDES Vol. 31, No. 9

Table 1. Values of the shell model parameters and electronic polarizabilities

Crystal y+ y_

O~f+ O~f _

k+ k_ (.~3) ()~3) (104 dyne/cm) (104 dyne/cm) (a) (a)

MgO 2.40 2.40 1415.3 34.3 0.094 3.88 CaO 4.35 2.40 929.9 34.3 0.47 3.88 SrO 4.95 2.40 650.5 34.3 0.87 3.88 BaO 5.65 2.40 475.7 34.3 1.55 3.88 SrS 4.95 4.35 650.5 42.8 0.87 10.2 BaS 5.65 4.35 475.7 42.8 1.55 10.2

(a) Reference [9].

Table 2. Values of input data, force constant and the exchange charge polarization parameter

r ~ ~Oro f Crystal (A) (10 -12 cm2/dyne) eo co. (1013 radian/sec) (104 dyne/cm) Die

MgO 2.102 a 0.645 b 9.8 e 2.94 h 7.462 i 10.55 - 0.436 CaO 2.406 a 0.8775 11.8 e 3.275 5.65 j 16.46 - 0.292 SrO 2.580 a 1.1375 13.3 e 3.275 4.276 k 13.62 - 0.323 BaO 2.762 a 1.6335 34.0 t 3.565 2.7521 10.15 --0.340 SrS 2.935 a 1.28 e 8.5 g 4.06 h 4.25 g 13.76 - 0.471 BaS 3.175 a 2.90 d 9.1 g 4.215 2.96 g 6.57 -- 0.769

(a) R. Wyckoff, Crystal Structures, Vol. 1. Wiley-Intersciences, New York (1961).

Co) C.E. Weir, J. Res. Nant. Bur Stand. 56, 187 (1956).

(e) M.L. Huggins & Y. Sakamoto, J. Phys. Soc. Japan 12, 241 (1957).

(a) P.W. Bridgman, Proc. Am. Acad. Arts Sci. 21, 74 (1940).

(e) B. Szigeti, Trans. Faraday Soc. 45, 155 (1949).

(O American Institute of Physics Handbook (1957).

0o L. Sirdeshmukh, Curt. Sc~ 43,712 (1974).

01) A.A. Winchell & H. Winchell, Optical Properties of Artificial Minerals, IIIrd ed. Academic Press, New York (1964).

(i) M.P. Verma & S.K. Agarwal, Phys. Rev. BS, 4880 (1973).

0) P.R. Vijayaraghavan & P.K. Iyengar, Proc. Int. Conf. (France) 139, (1971).

0o J.L. Jacobson & E.R. Nixon, J. Phys. Chem. Solids 29, 967 (1968).

O) M. Galtier, A. Montaner & G. Vidal, J. Phys. Chem. Solids 33, 2295 (1972).

ao, a** and e*[e quoted as experimental in Table 3 are obtained from the Clausius-Mossotti, the Lorentz- Lorenz and the Szigeti relations, which are

eo -- 1 4rr e o + 2 = ~ - ° t °

e** -- 1 4zr

3V e**+2

and 3CoTO [#V(eo-'eo~)] 'n

e*[e = ze(eoo + 2) [ 41r

where eo and e** are the low frequency and high fre-

quency dielectric constants,/a is the reduced mass per ion pair, and CoTO is the transverse optic mode frequency.

A comparison of the calculated and experimental values presented in Table 3 reveals that the shell model

(6) predicts correctly the values of e*[e. The shell model, however, predicts values of Oto and a** which are sub- stantially larger than the experimental values. It seems

(7) therefore necessary to take account of the exchange charge polarization as suggested by Dick and Overhauser [8]. In fact equations (2)-(4) have been derived by

(8) considering only the short range interaction polariza- tion mechanism. If the effect of exchange charge polari- zation is taken into account we obtain [8]

Vol. 31, No. 9 DIELECTRIC BEHAVIOUR OF ALKALINE EARTH OXIDES AND SULPHIDES

Table 3. Values o f low frequency and high frequency polarizabilities, and the effective charge parameter

665

~0 ~ (A 3) (A 3) e*/e

Exchange Exchange Shell Shell charge Shell charge

Crystal model Experimental ta) model model Experimental °~) model model Experimental (e)

MgO 5.15 3.31 2.64 1.86 1.74 0.58 0.44 CaO 6.72 5.20 3.27 2.72 2.86 0.64 0.54 SrO 8.60 6.59 3.88 3.28 3.53 0.70 0.59 BaO 12.05 9.22 4.83 4.20 4.63 0.78 0.65 SrS 11.41 8.62 8.99 7.56 6.10 0.52 0.34 BaS 19.88 11.15 10.71 8.28 7.90 0.75 0.42

0.60 0.64 0.63 0.76 0.58 0.49

(a) Derived from

Co) Derived from

(0 Derived from

equation (6).

equation (7).

equation (8).

Table 4. Average percentage deviations from the experimental values

Exchange charge Shell model model

a** + 27.2% + 2.40% e*/e + 8.86% -- 19.8%

O~o 1 1 1 (.v÷e -- D) 2

= (ze+D) 2 -=+. - -+- - + I k+ k_ k+

+ (Y_ek_+D)2+2(ze + D) [Y÷k+ D

O t . . - -

(.v÷e --D) 2 y+e --D

(y_e + D ) 2 [ k+ + k+ k+

(9)

+ D | 2 y_e k-]'

.y÷e -- D y_e + D]

e*/e = I + D + L k+ k_ I ze (~ _1+~_)

ze + k÷

1 + 1 1

f k÷ k_

(10)

(11)

where D is the exchange charge polarization parameter. For evaluating D from theoretical considerations we need to know the overlap integrals for nearest neighbour ions. In the lack of these integrals we can derive the values o f D phenomenologically using one of the equa- tions (9)--(11). We derive the values of D from equation (9) using experimental values of O~o. Values of Die so obtained are included in Table 2. Values of or** and e*/e

are then calculated and reported in Table 3. It is found that the inclusion of exchange charge polarization im- proves the agreement between theoretical and experi- mental values of c~** for MgO, SrS and BaS but for CaO, SrO and BaO the theoretical values oft~** are slightly smaller than the experimental values. Values of e*/e calculated by including the effect of exchange charge polarization turn out to be lower than the experimental values. Average percentage errors for ,~** and e*/e are given in Table 4. It is obvious that the inclusion of exchange charge polarization reduces the average per- centage error for t~**. For e*/e the situation is not im- proved. However, it is clear that the average percentage error is positive for the shell model and negative for the exchange charge model. Therefore an exact model for e*/e should be in between these two limits.

Finally it should be emphasized that we have for the first time studied six alkaline earth chalcogenides on the basis of the shell model employing the short range interaction polarization mechanism and the exchange charge polarization mechanism. The results reported in this paper may be useful for understanding the dielectric behaviour of other crystals of the family of alkaline earth chalcogenides.

REFERENCES

1. K.H. Rieder, B.A. Weinstain, M. Cordona & H. Bilz, Phys. Rev. B8, 4780 (1973).

2. G.R. Barsch & B.N.N. Achar, Phys. Status Solidi 35, 881 (1969).

3. R. Ruppin & R.W. Roberts, Phys. Rev. B3, 1406 (1971).

4. R. Srinivasan & K. Srinivasan, J. Phys. Chem. Solids 33, 1079 (1972).

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5.

6. 7.

DIELECTRIC BEHAVIOUR OF ALKALINE EARTH OXIDES AND SULPHIDES Vol. 31, No. 9

M.J.L. Sangster, Solid State Commun. 18, 67 (1976). E.E. Havinga, Phys. Rev. 119, 1193 (1960). B.G. Dick, Phys. Rev. 145,609 (1966).

8. B.G. Dick & A.W. Overhauser, Phys. Rev. 112, 90 (1958).

9. L. Pauling, Proc. Roy. Soc. All4, 181 (1927).