THE DIELECTRIC RESPONSE OF CRYSTALS

(1)

HAL Id: jpa-00214951

https://hal.archives-ouvertes.fr/jpa-00214951

Submitted on 1 Jan 1972

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

THE DIELECTRIC RESPONSE OF CRYSTALS

G. Coombs, R. Cowley

To cite this version:

G. Coombs, R. Cowley. THE DIELECTRIC RESPONSE OF CRYSTALS. Journal de Physique Colloques, 1972, 33 (C2), pp.C2-57-C2-58. �10.1051/jphyscol:1972215�. �jpa-00214951�

(2)

JOURNAL DE PHYSIQUE Colloque C2, supplgment au no 4, Tome 33, Avril 1972, page C2-57

THE DIELECTRIC RESPONSE OF CRYSTALS

G. J. COOMBS and R. A. COWLEY Physics Department, University of Edinburgh, Scotland

Rksum6. - En considerant exactement le couplage entre les fluctuations ferroelectriques, et les fluctuations B basses frequences de la densite de phonons, d'un cristal piezoClectrique, nous avons obtenu une nouvelle contribution a l'energie propre des phonons. Nous avons suppose que ces termes rendaient compte d'une dependance anormale du temps de relaxation en fonction de la temperature pour le mode ferroelectrique. Cette dependance anormale a kt6 observke lors de mesures de dispersions de lumikre. La modification resultante dans la fonction de dispersion est aussi considere.

Abstract. - By properly considering the coupling of the ferroelectric fluctuations to the low frequency fluctuations in the phonon density of a piezoelectric crystal, a new contribution to the phonon self-energy is obtained. It is suggested that this terms accounts for the anomalous temperature dependence of the relaxation time of the ferroelectric mode that has been observed in recent light scattering experiments. The resulting change in the scattering function is also considered.

In considering the low frequency dielectric response of a crystal, care must be taken in the treatment of the fluctuations in the phonon density to which the light couples through the transverse optic phonon. If the time scale of the fluctuations is long compared with some average phonon lifetime then the probability of phonon decay in the intermediate virtual states must be taken into account. We shall show that these consi- derations lead to a new contribution to the phonon self-energy of a piezoelectric crystal. The contribution is absent in crystals of higher symmetry as the relevant coupling coefficient is identically zero. In ferroelectric materials, it is obvious that this contribution will be important due to the large increase in the amplitude of the ferroelectric fluctuations as the phase transition is approached.

Kubo [I] has shown that the first order response function of a system is just the one-particle retarded Green's function. For the weakly anharmonic lattice dynamical model of a dielectric crystal, we may write the response function of the mode with wave vector q belonging to branch j as

G-'(qi, o ) = 02(qj) - o2

+

~ ( q j , a )

.

(I) Here w(qj) is the harmonic phonon frequency of this mode and ~ ( q j , o ) is the self-energy of the mode due to the anharmonic terms in the Hamiltonian of the crystal. If we consider only the cubic anharmonic term then the lowest order perturbation theory contribution to the self-energy may, as was first demonstrated by Maradudin and Fein [2], be represented by the diagram shown in figure 1. The self-energy is a complex function of frequency : the real part making a small correction to the phonon frequency which we absorb into o(qj) to give the shifted frequency G(qj) It is the imaginary part of this function that is of interest. For the high frequency response it is sufficient to treat the internal lines of the diagram in the harmo-

FIG. 1. - Diagram representing the lowest order perturbation expansion contribution to the phonon self-energy. The coupling

coefficient is associated ^withone of the vertices.

nic approximation and the imaginary part so obtained may be approximately represented by the first term in its Taylor series expansion which we write as

+

2 y(qj) o. The response function is now given by

which is the well-known result for a classical damped oscillator. I t has been shown by Yamada, Shirane and Linz [3] that this form gives a reasonable description of the response of the soft transverse optic mode in non-piezoelectric BaTiO,. Cowley et al. [4] have demonstrated that this form is not adequate for the description of the light scattering from the piezoelectric materials CsH,AsO,(CsDA) and KH,AsO, (KDA) which undergo ferroelectric transitions. To correctly treat the coupling of the dielectric fluctuations to the low frequency fluctuations in the phonon density, we must revaluate the diagram of figure 1 ; replacing the harmonic phonon lines by thermal phonon lines for which we use the form of eq. 2. In the low frequency limit the predominant contribution ot the phonon self-energy occurs when the two phonons represented by the internal lines of the diagrams are of the same branch. This was first pointed out by Sham [5]. The anharmonic coefficient which gives the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972215

(3)

C2-58 G. J. COOMBS AND R. A. COWLEY coupling of the q ⁼0 transverse optic phonon to

those two phonons may be written as

and it is this coefficient that is identically zero for non-piezoelectric crystals.

Using techniques similar to those developed by Sham [5] and Klein and Wehner [6], the resulting contribution to the self-energy may be evaluated.

There are two cases of interest. In the first case of piezoelectric but not pyroelectric crystal, the transverse optic mode couples to phonon density fluctuations which represent only deviations from the local thermodynamic equilibrium. In the second case of pyroelectric crystals, the transverse optic modes couples, in addition, to the fluctuations in the local thermodynamic equilibrium. In the former case the self-energy becomes

aT 1

+

i o z

where a is a positive constant and z is strictly a decay time for the deviation from local thermodynamic equilibrium but may be taken as some average phonon lifetime. We have made the high temperature expansion of phonon occupation factors with the result that the term is linearly dependent on the temperature T. The remainder of the self-energy has the same form as that discussed above. The response function for the ferroelectric mode is then

2 aT

G-'(W) =

iZi

- o

+

iwy - ---

1

+

i o z (3) whereas at high frequencies the reponse of the crystal was determined by the soft mode

o; = k(T

-

To)

where k and To are constants, the response is now determined by

-2 oo - a T .

This follows if we note that the dielectric susceptibility is just proportional to this reponse function and if we assume that the static susceptibility has the usual experimentally determined form

- 1

Xo aT - Tc

where T, is the ferroelectric transition temperature.

We see that

T - - o - (k

i ;").

which implies that

T, > To

.

It is this difference between the high and low frequency response that hes been observed in the Raman scattering measurements on CsDA and KDA [I].

It is the imaginary part of the response function which determines the scattering properties of the crystal. The nature of the scattering described by the function we have derived depends sensitively on the parameters of the model. In some cases there are two peaks in the scattering which coalesce into one if either the temperature is raised or the strength of the an harmonicity parameter is increased. Unfortunately detailed experimental masurements are not available in the region o~

-

1, and so we are not able to compare the results of these calculations with experiment in detail.

For relatively large values of the anharmonicity parameter a and temperatures not too chose to the transition the low frequency part of the scattering function may be approximately written as

where

and

This just the Debye relaxation form with a relaxation time that diverges as the transition is approached.

This form of the response, including the effects of different relaxation times, has been discussed in connec- tion with a number of compounds by Yoshimitsu and Matsubara [7].

In pyroelectric crystals the form of the self-energy becomes more complicated but has qualitatively a similar structure. We expect that at small o there will be an anomalous contribution to the self-energy which will give rise to additional response at low frequencies. The new features which occur in pyroelectric crystals are that the form of the response is singular as q and w --+ 0 and that the transverse optic mode may also couple to a second sound mode. We intend to publish a more detailed account of these results in the near future.

References

[I] KUBO (R.), J. Phys. Soc. Japan, 1957, 12, 570. RYAN (J. F.) and SCOTT (J. F.), J. Phys. C. Solid [2] MARADUDIN (A. A.) and FEIN (A. E.), P h y ~ Rev., ^StPhys., 1971, 4 , L 185.

1962, 128, 2589. [5] SHAM (L. J.), Phys. Rev., 1967, 156, 494.

[6] KLEIN (R.) and WEHNER (R. K.), Phys kondens.

[3] YAMADA (Y.), SHIRANE (G.) and LINZ (A.), Phys. Rev., Materie, 1961, 10, 1 .

1969, 177, 848. 171

. -

YOS~MITSU (K.) and MATSUBARA (T.), pro^. Theor.

[4] COWLEY (R. A.), COOMBS (G. J.), KATIYAR (R. S.), ^Phys., 'Supplement Extra

umber

^1968,^109.