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ON THE MICROTHEORY OF SPONTANEOUS POLARIZATION AND PHASE TRANSITIONS IN CRYSTALS OF PEROVSKITETYPE AND OF ONES
WITH TETRAHEDRAL STRUCTURAL UNITS
I. Bersuker, B. Vekhter, A. Muzalevskii
To cite this version:
I. Bersuker, B. Vekhter, A. Muzalevskii. ON THE MICROTHEORY OF SPONTANEOUS POLAR- IZATION AND PHASE TRANSITIONS IN CRYSTALS OF PEROVSKITETYPE AND OF ONES WITH TETRAHEDRAL STRUCTURAL UNITS. Journal de Physique Colloques, 1972, 33 (C2), pp.C2-139-C2-140. �10.1051/jphyscol:1972246�. �jpa-00214982�
JOURNAL DE PHYSIQUE Colloque C2, supplkment au no 4, Tome 33, Avril 1972, page C2-139
ON THE MICROTHEORY OF SPONTANEOUS POLARIZATION AND PHASE TRANSITIONS IN CRYSTALS OF PEROVSKITETYPE
AND OF ONES WITH TETRAHEDRAL STRUCTURAL UNITS
I. B. BERSUKER, B. G. VEKHTER and A. A. MUZALEVSKII Department of Quantum Chemistry, Academy of Sciences of Moldavian SSR,
Kishinev, 28, USSR
Resumk. - Un bref aperw sur la microthhrie de la ferroelectricitt en cours de developpement resume ses propriktts fondamentales et les idees de base : parmi les points fondamentaux de cette theorie on trouvera une explication de la polarisation spontank et I'origine des transitions de phase, pour une grande varittk de cristaux. Des exemples du type perovskite (BaTi03), du type sel de roche (GeTe) ainsi que d'autres types tels TGS et KDP sont Btudies ; les rksultats sont compatibles qualitativement avec les experiences.
Abstract. - A breaf outlook of the developing microtheory of ferroelectricity is presented summarizing its main features and the basic physical ideas. In the framework of this theory an explanation of the spontaneous polarization and phase transitions origin in a wide range of different type of crystals is given. As examples the perovskite type BaTiO3 and rock salt type GeTeas wellas TGS and KDP type crystals are considered, the result being in a good qualitative agreement with the experiments.
In the theory of ferroelectricity presented [I]-[6] we start from the exact Hamiltonian by means of which in the adiabatic approximation the electronic structure for fixed nuclei and the nuclear adiabatic potential are determined at first and then the dynamics of nuclear motions under this potential is considered. By this approach the origin of the strongly anharmonic potential needed for the ferroelectricity origin expla- nation is due to the interaction which mixes the ground and excited electronic states. If the mixing is strong enough, i. e. if the condition
is satisfied, where a is the vibronic coupling constant for the interacting states mixed by the dipole type nuclear displacements, k is the force constant and A is the energy difference of the electronic states under consideration for the symmetrical (undisplaced) nuclear configuration, the later becomes unstable as regard the dipole type displacements mentioned above (dipole instability [I]).
For cubic ionic crystals of perovskite (BaTiO,) and rock salt (GeTe) types the interacting states are the valence and conductivity bands and the dipole type ferroelectric displacements are the three-fold degenerate optical (q = 0) vibrations (Q,, Q, and Q, coordinates) [I], [2]. If the condition similar to (1) (affected by the account of dispersion) is satisfied the adiabatic potential &(ex, Q,, QJ has the following extrema points : eight mnima at
for which the crystal is polarized in the trigonal axes directions, two types of saddle points-12 at
and 6 at
the later being higher in energy, and a maximum at all Q, = 0
('1.
For GeTe type crystals the conditions of different type extrema realization are some diffe- rent so a situation is possible when only 8 trigonal minima and a saddle point at all Q, = 0 occur.At T = 0 such a crystal is polarized along the tri- gonal axes, the magnitude of the polarization being depended on the crystal parameters in (1). At higher temperatures phase transitions in some averaged on different minima direction states may occur, the later being determined by the adiabatic potential. The mentioned above E(Q) peculiarities for BaTiO, lead to the observed experimentally three phase transitions ; GeTe may have only one phase transition. In a semi- classical approach the equations connecting the Curie temperatures T, and the theory constants A , a, k can also be obtained [4] :
For BaTiO, two Curie temperatures can beyused for estimation of two parameters, A and y, and then the
(1) Only the linear in Q approximation is accounted for.
The second order terms lead to a renormalization of k [2]
which as it was shown for BaTiO3 gives no instability [3].
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972246
C2-140 I. B. BERSUKER, B. G. VEKHTER AND A. A. MUZALEVSKII
third T, can be calculated, T,(~' = 201 OK, while the where 52 is the frequency due to tunneling between the experimental value is T,(~) = 183 OK. The calculation minima states on the same site (the q,, are not ortho- based on (2) gives a good qualitative agreement with gonal). The second term in (3) discribes the tetrahedra the experimental data ; the following examples are interactions. Using RPA decoupling for the Green
illustrative : function method one can obtain the transcendental
equation that determines the phase transition tempera- a) Tc(BaTi03) > Tc(BaZr03) > Tc(BaHf03) ture. In the approximation I
+
52 we have :and
Tc(BaTiO,) > Tc(SrTi03) > Tc(CaTi03) ; b) for rock salt type crystals
c) for TGS type crystals :
One of the features of this result is of special impor- tance. As far as the adiabatic potential minima are trigonal ones, only the rombohedral phase is comple- tely ordered, the others are some mixtures of different oriented trigonal distorted elements. In the semiclas- sical consideration mentioned above [4] the detailed structure of these not completely ordered phases remains uncertain as the vibration dispersion acount in a more complete investigation is rather difficult and is not drawn out yet. The experimental results of Guinier, Comes and Lambert [7] seems to support this general statement giving an impressive picture of the microstructure of these phases.
Another great number of ferroelectries is that containing tetrahedral structural clusters of XY, type.
As it was shown [5] the XY, system under the condi- tion (1) possesses four equivalent minima of the adia- batic potential with a dipole moment along one of the X-Y links in each of the minima. In the secondary quantization representation on the basis of qm,- functions, which discribe distorted tetrahedra (in the y-th minimum of the m-th lattice site) the Hamilto- nian may be written as follows :
-
C
W,,,(mmr) n,, n,.~r. (3)mm' p ~ '
A distinguished example provides the KDP crystal which in addition to the considered above 52 tunneling on the tetrahedra PO, has a proton tunneling in the hydrogen bond essentially affecting the ferroelectricity properties as it is seen from the isotope effect. It can be taken into account by means of an additional term in (3), o
C C
b; b,,,, where o is the proton1 xx'
tunneling frequency, and in the same approximation as for (4) one can obtain the following relation for the Curie temperature [6] :
where I and I, are some combinations of tetrahedron- proton interaction constants. As the tunneling fre- quencies 52 and o essentially depend upon the atomic masses, equation (5) shows the possibility of an addi- tional isotope effect on the tetrahedron too which may occur by the oxygen and phosphor isotope substitu- tion. The experiments of Wiener, Levin and Pelah [8]
may be regarded as supporting this result.
Estimating of the I and I, parameters, say in the approximation of only neighbouring dipole-dipole interaction, one can draw out some rough values for the T,. For the KD,PO, and KH,PO, cases we have obtained
T,(~) = 114 OK and
T,(H)
= 74 OKrespectively which are lower than the experimental data, but the
T,(D)/T,(H'
= 1.6 is very close to the later (1.7). The account of not only the neighbouring interaction will evidently increase the I and I, and consequently the T,.References [I] BEESUKER (I. B.), Phys. Lett., 1966, 20, 589 ; Teor. i
exp. chim. 1969, 5, 293 ; )MI1 IUPAC Congress, Sydney, 1969, Abstracts, 16 ; (( Stroenie i svoistva coordinatsionnih soedinenii. Vvedenie v teoriu )), Chimia, 1971, str. 116.
[2] BERSUKER (I. B.), VEKHTER: B. G.), Fiz. Tverd. tela, 1967, 9 , 2652 ; I1 Intern. meeting on Ferroelec- tricity, Japan, 1969, Abstracts, 274.
[3] BERSUKER (I. B.), VEKHTER (B. G.), MUZALEV- sKn (A. A.), Zzv. AN MSSR ser. biol. i chim.
nauk, 1969, 4, 70.
[4] BERSUKER (I. B.), VEKHTER (B. G.), IZV. Akad. Nauk SSSR, Ser. jiz. 1969, 33, 199.
[5] BERSUKER (I. B.), VEKHTER (B. G.), DANILCHUK (G. S.), KREMENCHUGSKII (L. S.), MUZALEVSKII (A. A.) and RAFALOVICH (M. L.), Fiz. tverd. tela, 1969, 11, 2452.
[6] BERSUKER (I. B.), VEKHTER (B. G.), MUZALEVSKII (A.
A.), Phys. Stat. Sol. (b), 1971, 45, K 25.
[71 COMES (R.), LAMBERT (M.), GUINIER (A.), Sol. St. Cum.
- -
1968, 6, 715.
[8] WIENER (E.), LEVIN (S.) and PELAH (I.), J. Chem. Phys.,
