HAL Id: jpa-00214955
https://hal.archives-ouvertes.fr/jpa-00214955
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 pub- lished 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.
INFORMATION ABOUT THE PHONON SPECTRUM OF FERROELECTRIC OBTAINED BY LIMITATION
OF CRYSTAL SIZE
E. Bursian, Ya. Girshberg
To cite this version:
E. Bursian, Ya. Girshberg. INFORMATION ABOUT THE PHONON SPECTRUM OF FERRO- ELECTRIC OBTAINED BY LIMITATION OF CRYSTAL SIZE. Journal de Physique Colloques, 1972, 33 (C2), pp.C2-69-C2-71. �10.1051/jphyscol:1972219�. �jpa-00214955�
JOURNAL DE PHYSIQUE Colloque C2, supplkment au no 4, Tome 33, Avril 1972, page C2-69
INFORMATION ABOUT THE PHONON SPECTRUM
OF FERROELECTRIC OBTAINED BY LIMITATION OF CRYSTAL SIZE
E. V. BURSIAN and Ya. G. GIRSHBERG
Laboratory on ferroelectricity Gertsen State Pedagogical Institute Leningrad, USSR
R6sum6. - On a supposk que la variation des propri6tes diklectriques des films minces ferro- 6lectriques avec I'kpaisseur pouvait donner des informations sur la valeur de B dans la formule de dispersion w $ = A(T - To) + Bk2, ce qui n'est le cas ni pour la diffusion de neutrons et ni pour les mkthodes optiques.
Abstract. - The supposition is advanced, that the measuring of thikness dependence of a ferro- electric thin films dielectric properties may give an information about the B value in the formula
w; = A(T - To) + BkZ in diapason, usually not empbraced by neutron scattering and optic methods.
In reference [I] the authors, investigating the dependence of on the thikness of crystal, suggested the steep slope of the soft mode o,(k) at small k in the region above the transition. This is confirmed by the direct measurings on microwaves [2], where a considerable difference between the soft mode o, and the neutron scattering data [3] is obtained. Apparently this difference accounts for the fact that the neutron scattering gives the information about phonon spec- trum for k from 10' up to lo7 cm-', while studying the dispersion of electromagnetic waves (microwaves and IR-spectroscopy) we obtain the information
FIG. 1. - Qualitative picture of the dispersion curves character.
The Region I - polariton range, region I1 - the range of k
discussed in communication, region I11 - the fundamental Brilluen zone part. The scale of various region is different.
about o(k) in the region 10,-lo4 cm-I. The special important and interesting intermediate region of small k is not empbraced by these methods (Fig. 1).
The supposition about the possibility of investiga- tion this very diapason is advanced here. Restriction of the crystal dimension in some direction z up to d leads to the dividing the Brilluan zone into layers, so that only kZ = m z/d can be realized, where m = 1,2.. .
The minimum value of k, is no longer equals zero, but n/d. AS it was shown in (41, the functional depen- dence o ( k ) is not changed (here and below the effect of delay is not taken into account, the delay is essential at k < lo4 cm-l). The limitation of k in its turn leads to the restricting the possible minimal values of the frequency o > omin = f (d). This modification don't affect the modes, which have no peculiarities in the k = 0 region, but influence strongly a soft mode.
Taking this into consideration as well as Lidden- Sakse-Teller (LST) relation one can obtain
Hence, the detailed investigation of eo(d) allowes to obtain the dependence o,(k) at small k.
In spite of the rough approximation of the one dimension chain of anharmonic interracted ions in Silverman model [5], this approximation allowed to explain successfully microwave losses in SrTiO,, so that the Silverman Hamiltonian may be considered to have an essential characteristics of the ferroelectrics with perovskite structure. In [6] it was shown that in this model
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972219
C2-70 E. V. BURSIAN AND YA. G. GIRSHBERG
where B is a positive, k-independent constant (Fig. 2).
Using (2), we obtain
Unfortunately, the graph c0(4 obtained in [I]
evaporation of electrodes on a degassed surface in high vacuum, dL/sL x 10 181.
At d-I > 4 x lo3 cm-l experimental points lie below the straight line. It allows to make a rough preliminary estimation of ATo2 and therefore B.
Apparently
-
lo8 cm/s, that is it's larger than the sound velocity of the order of 1 t 2.As we see, the proposed method is essentially based on two suppositions :
1. The soft mode o,(k) has a remarkable dispersion which is a result of anharmonicity accounting (Sil- verman model).
2. The LST-relations hold good for anharmonic crystalls too.
The latter is not obvious. LST can be obtained from phenomenological equations [9] :
FIG. 2. - Qualitative image o ~ ( k ) in the region I1 at different temperatures. 1 - T = To, 2 + 6-the increase of temperature.
cannot be utilized directly for B definition, since the surface layers of different nature influence the E,
effective value. The characteristic indication of this influence is the E~ frequency dependence in region
< lo5 S-1 171.
Let both phenomena take place. Then
1 1 2 d, C BZ' C
-- = - [(T - To) + -
80 eff C d ~ , d2 E, 0 2
where d,, eL are the surface layers thikness and die- lectric susceptibility. Thus the Curie-Weiss law is not disturbed, the constant C remaines the same, and both phenomena lead to appearance of two additive correc- tions to To, so in the experiment we detect on the plane (&-I, T) a parallel shift of the straight line upwards. (It should be noted, that the measuring of To ,,,(d) is a more exact method, than the measuring of e0 eff(d) for an error in determining crystal geome- trical sizes does not tell on the result.)
Owing to different functional dependence on d these corrections may be separated (at large d only the first correction is essential). If one draws a straight line across the experimental points on the plane (To .,,, d - l ) in a region of small d-l, it is possible to find ATo,(d) and dL/eL. According to preliminary data obtained by
Here w is an ion displacement, E and P-macroscopi- cal field and polarization. The system (5) is a harmo- nic approximation and in this case the coefficients bik do not depend on co-ordinates and time and are connected by ordinary relations with the measurable values e0 and E, [9]. It is clear, such consideration needs a revision, since the first supposition is based on anharmonicity accounting whereas the second one ignores it completely.
Let us take the anharmonicity into account indi- rectly supposing the equations (5) to be just, but the coefficients bik to be dependent on co-ordinates. From symmetry consideration it is clear, bik may depend only on difference of co-ordinates, that is
The standart expression for P(r), which does not take into account a space dispersion, follows from (61, if bik(r - rf) = bi;) 6(r - rr).
The obtained integral equations are solved easily by a transition P(r) and E(r) to Fourier-form :
P(k, o ) =
1
P(r, t) e x p ( ikr + iot) dr dt (7) E(k, o ) =1
E(r, t) exp(- ikr + iot) dr dt .Using (6), (7) we have
INFORMATION ABOUT THE PHONON SPECTRUM C2-71 Solving the system (8) we find ~ ( k , o) from the relation Introducing ~ ( k , 0 ) = ~ , ( k ) and ~ ( k , a) = &,(k) we
of P(k, w ) and E(k, a) : obtain LST-relation in presence of any dispersion :
So we obtain E , depending not only on o as usual, but o n k too.
As it is known [ l o ] , roots of ~ ( k , w ) correspond to the longitudinal oscillations o, and poles correspond t o the transversal ones (w,). From (9) we obtain
Of course, in the experiments with thin films the practically measured dielectric susceptibility is not
~ ~ ( k ) but it is some E ~ Nevertheless one may use ~ ~ ~ .
for the estimation ~ , ( k ) a t kmin = n/d instead of E , ,,,,
for the main deposit in susceptibility gives only the smallest k . Non-dependencity (a weak dependencity)
of o, and E , on k is supposed.
So the initial supposition about a possibility of transverse mode dispersion wT(k) investigation by crystal size limitation is correct.
References [I] BURSIAN (E. V.), GIRSHBERG (Ya. G.), MAKAROV
(K. V.), SAIKOVSKII ( 0 . I.), Fis. Tverd. Tela (Sov. Phys. Solid State), 1970, 12, 1850.
[2] BURSIAN (E. V.), RYCHGORSKI (V. V.), GIRSHBERG (Ya. G.), Fis. Tverd. Tela (Sov. Phys. Solid State), 1971, 13, 541.
[3] YAMADA (Y.), SHIRANE (G.), LINZ (A.), Phys. Rev., 1969, 177, 848.
[4] F u c ~ s (R.), KLIEWER (K.), Phys. Rev., 1965, 140, A 2076.
[5] SILVERMAN (B.), Phys. Rev., 1964. 125, 1921.
[6] TAN1 (T.), J. Phys. Soc. Jap., 1969, 26, 113.
I71 SHLOSSER (H.), DROUGARD (M.), J. Appl. Phys., 1961,
- . - -
32, 1227.
[8] BURSIAN (E. V.), GIRSHBERG (Ya. G.), MAKAROV (K. V.). Abstracts of the VII scientific conference on fer;belectricity, USSR, Voronezh, 1970, p. 120.
[9] BORN (M.), HUANG (K.), Dynamical Theory of Crystal Lattices (Oxford University Press, New York, 1954).
[lo] PINES (D.), Elementary excitations in solids. (W. A.
Benjamin, inc. New York - Amsterdam, 1963).
