MICROSCOPIC POLARIZABILITY MODEL OF FERROELECTRIC SOFT MODES

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MICROSCOPIC POLARIZABILITY MODEL OF

FERROELECTRIC SOFT MODES

A. Bussmann-Holder, G. Benedek, H. Bilz, B. Mokross

To cite this version:

JOURNAL DE PHYSIQUE

CoZZoque C6, suppZ6ment au n012, Tome 42, dgcembre 1981 page c6-409

MICROSCOPIC POLARIZABILITY MODEL OF FERROELECTRIC S O F T MODES A. Bussmann-Holder, G. ~enedek*, H. Bilz and B. Mokross

Max-PZanck-Institut fiir Festkiiperforschwzg, 7000 Stuttgart-80, F.R.G. *~stituto di Fisica deZZt ~niuersi& and CRSM-CNR, MiZano, Italy

Abstract.- A simplified veryion of a recent microscopic model of ferroelectric soft modes is studied. It is sh2wn that the compensation of long-range and short-range forces which indu- ces the soft mode behaviour of ferroelectric systems can be ex- pressed in terms of a simple linear chain model with a ngn-li- near polarizability at the chalcogenide ion lattice site

.

This polarizability is equivalent to an on-site electron-two-phonon coupling. The four different temperature regimes which result from the model equations are discussed. The model is

applicable

to coqjpletely different systems g u ~ h as perovskites, SbSI

,

IV-VI semiconductors and K2Se04

'

.

The dynamical properties of ferroelectric perovskites have been successfully described in terms of a strongly anisotropic non-linear polarizability of the oxygen ion'. By means of a simplified diatomic linear-chain version of this model8 it was possible to describe as well the soft mode properties of other systems and to interpret the carrier9 and defect1' concentration dependence of the soft mode in terms of microscopic parameters. Within the model the instability of the ferroelectric soft mode is attributed to a negative electron-ion coupling constant g2 which contains strongly attractive Coulomb for- ces in a local approximation. Stabilization of the paraelectric modes is guaranteed by the repulsive on-site fourth-order electron-ion coup- ling g4 and the second nearest neighbour coupling £'.

The model has some features in common with those discussed by Pytte et a1.I1 as it also represents a double-well problem, where the finite electron core displacement w is analogous to the static ionic displacements. The classical ground state is given by wz =

-

g2/g4. 1 2 It is however important to note that in this model the double well po- tential results from the non-linear interaction between electrons and ions. Similar to the double well problem a border line for displacive and order-disorder regimes may be defined which is given by

92 f <

2 (=) ( 1

+

F ) 1 l 2 where the second nearest neighbour repulsive Coulomb interaction f' is essential for the existence of the displaci- ve regime.

JOURNAL DE PHYSIQUE

Within the model the tem- perature dependence of the fer-

+Mijller et al. -

r

x Hach\i , ~ y t z

w:-€:-(T-T~)

.Balkanski et aC

roelectric soft mode is given by p = reduced mass 2 2fg f = nearest ''Wf = 2fig neighbour core-shell- coupling A Z where g(T) = g2

+

3g4 <w > 2

and <w > is the self-consistent thermal average over the squa- red relative core-shell displa- cements at temperature T.

The different temperature regimes which result from the model are shown in Fig. 1. For

high enough temperatures the soft mode saturates and the cri-

2 tical exponent y with wf

=

( T - T ~ ) ~

,

reaches 1/3 of its Fig. 1 . Temperature regimes which mean-field value 1. Experimen- result out of the model, exp.: _{tally this has been observed}

+

14), X 15)r 4 ) .

for SbSI , 4 KTa03 and SrTi03. For finite Tc and temperature T>Tc the well-known mean-field regime occurs which governs the largest temperature region. y=l has been ob- served for all ferroelectric systems. For Tc:O and temperatures close t C O K again strong deviations from y=l are observed. In the quantum limit classical renormalization group theory predicts an unchanged ex- ponent y=l. The quantum limit leads to an apparent enhancement of dimensionality, i.e. d=4, which yields for the critical exponent: y=2

(refer to D. Rytz, U.T. HBchli, this Conference). The self-consistent phonon approximation (SPA) which leads to exact results in the very displacive limit predicts y=2, too13.The dimensionality crossover re- gime, appearing at low temperatures, is dominated by a "crossover" expo- nent, 1.4<y<1.6. All temperature regi-mes have been observed experimen- tally as indicated in Fig. 1.

(refer to H. Biittner, this Conference).

A more extended version of the SPA and the non-linear problem 13,6

will be given elsewhere

.

References

1. R. Migoni, H. Bilz, D. Bauerle, Phys. Rev. Lett.

37,

1155 (1976)

2. P.W. Anderson, Fizika dielectrikov, 1959 ed. G.J. Skanavi (Akad. Nauk SSSR, Moscow) p. 290, !?1. Cochran, Adv. in Physics

9 ,

387

( 1 960)

3. A. Bussmann, H. Bilz, R. Roenspiess, K. Schwarz, Ferroelectrics

25, 343 (1980)

-

4. M. Balkanski, M.K. Teng, M. Massot, H. Bilz, Ferroelectrics

3,

737 (1980)

5. A. Bussmann-Holder, H. Bilz, W. Kress, Proc. 1 5Int. Conf. ~ ~ Physics of Semiconductors, Kyoto 1980,

J. Phys. Soc. Japan

49,

A737 (1980)

6. H. Bilz, H. BUttner, A. Bussmann-Holder, W. Kress, U. SchrBder,

to be published

7. A. Bussmann-Holder, H. Bilz, H. Biittner, Ferroelectrics, to be published

8. H. Bilz, A. Bussmann, G. Benedek, H. Biittner, D. Strauch, Ferroelectrics

25,

339 (1980)

9. A. Bussmann-Holder, H. Bilz, bl. Kress, U. SchrGder, to be published

10. A. Bussmann-Holder, H. BiLz, D. Bauerle, D. Wagner, 2 . Phys. B 41 353 (1981)

- 1

1 1 . E. Pytte, Phys. Rev. B

5,

3758 (1972)

12. H. BUttner, H. Bilz, in Recent Developments in Condensed Matter Physics Vol. 1 (ed. Devreese) p. 49, Plenum (1981)

13. H. Bilz, A. Bussmann-Holder, G. Benedek, H. Biittner, D. Strauch, to be published in 2 . Phys.