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BORACITES - AN EXAMPLE OF IMPROPER FERROELECTRICS
V. Dvo_ák
To cite this version:
V. Dvo_ák. BORACITES - AN EXAMPLE OF IMPROPER FERROELECTRICS. Journal de Physique Colloques, 1972, 33 (C2), pp.C2-89-C2-90. �10.1051/jphyscol:1972226�. �jpa-00214962�
JOURNAL DE PHYSIQUE ColIoque C2, supplkment au no 4, Tome 33, Avril 1972, page C2-89
BORACITES - AN EXAMPLE OF IMPROPER FERROELECTRIC S
Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia
Rbsum6. - Une classification des ferroblectriques est decrite brikvement. Les ferroblectriques impropres dans lesquels par definition, le parambtre de transition n'est pas la polarisation, sont classes selon le vecteur k des paramares de la transition et de la forme de l'energie #interaction donnant la polarisation spontanee. Quelques proprietes des boracites (qui sont representatifs d'un groupe de ferroelectriques impropres) pres de leur transition cubique-orthorhombique sont donnbes.
Abstract. - A classification of ferroelectrics is briefly discussed. Improper ferroelectrics, in which by definition the phase transition parameer is not the polarization, are classified according to the k-vector of $he phase transition parameters and the form of interaction energy leading to the spontaneous polarization. Some properties of boracites (which are representatives of one group of improper ferroelectrics) near their cubic-orthorhombic phase transition are described.
The group-theoretical formulation of Landau theory of phase transitions provides a natural basis for classification of structural phase transitions (PT) in crystals. Knowing the symmetry change of a crystal one can find the irreducible representation which induces the PT and accoring to which the PT parame- ters transform [I]. Then using simple rules of the representation theory it is possible to deduce which macroscopic properties should change due to this PT.
Indenbom [2] has applied this procedure to some equitranslational (no sublattices occur in the low- symmetrical phase) PT. His paper, however, has not been fully appreciated and instead of completing this analysis for all possible point group changes and generalizing it also for non-equitranslational PT, the Aizu's classification [3] is now frequently used. This classification concerns equitranslational PT only and actually it is a special case of a more systematic classi- fication (not done yet !) based on the group represen- tation theory.
Basically two types of ferroelectric PT can be dis- tinguished : proper, in which the homogeneous pola- rization P is the parameter of PT, and improper, in which it is not. For brevity we will use the terms proper and improper for a ferroelectric crystal itself too.
Strictly speaking this is sometimes ambiguous since the same crystal can display different (proper or improper) ferroelectric PT but we believe it will lead to no misunderstanding.
Essential characteristics of proper ferroelectrics (PF) (BaTiO,, TGS, SbSI) are : Curie-Weiss law for the dielectric constant in the paraelectric phase, no sub- lattice formation in the ferroelectric phase. As pointed out first by Indenbom [2], P, in improper ferroelectrics (IF) arises due to coupling of P with PT parameters.
Therefore we will classify I F according to the type of this coupling. We will consider only two lowest order
interaction terms of the form pP and p<P [4] where p and 5 are PT parameters. (It should be pointed out that interaction terms containing P2 could in principle lead to P, too, provided non-linear polarization energy
- P4 is taken into account [5]. No I F of this type is known so far.) I F can be divided into two large groups :
A) PT parameters are homogeneous (k = 0) - no sublattices occur.
B) PT parameters are non-homogeneous (k # 0) - sublattice formation. (The concept of equitransla- tional and non-equitranslational PT was introduced by Indenbom [2] and also by Ascher [6].) The group A can be further subdivided according to which inter- action term allowed by the crystal symmetry is the lowest order one : A 1) pP and A 2) PCP. Clearly, the interaction term pP can exist only if p transforms like P [4]. In this case two groups can be further distin- guished :
A la) p is a strain component. This is possible in piezoelectric classes only and essential characteristics of such PT would be : cP, cE (elastic constant at constant P and electric field E) and free c show pronounced temperature anomalies in the paraelectric phase.
A lb) p transforms like P but the corresponding conjugate force cannot be controlled experimentally.
The macroscopic properties of such I F are very similar to those of P F except the different role of the depola- rizing energy in the ferroelectric phase [4]. KH,PO, seems to be an example of such I F and p in this case characterizes the ordering of protons in hydrogen bonds [7].
Representatives of the group A 2 could be found in piezoelectric classes C,,, D,, and T, only [2]. No temperature anomalies of any material constants in the paraelectric phase and no new sublattices would be observed in this case.
7
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972226
C2-90 V. DVORAK
FinalIy let us discuss the group B characterized by sublattice formation, interaction y t P and hence by no macroscopic anomalies in the paraelectric phase. Both piezoelectric (Gd2(Mo04), [8] and nonpiezoelectric ((NH4),BeF4 [4]) examples are already known.
Cubic-orthorhombic (T:-C;,) PT in boracites (Me,B,O,,X) provides another example of an I F PT belonging to the group B. Group-theoretical analysis showed [9] that this PT should be described by two parameters 5, and 5, with the wave vector at the BZ corner, i. e. k = t(bl i- b2). (Therefore this PT is connected with the doubling of the primitive unit cell.) Actually these two parameters belong to a six dimen- sional irreducible representation of T: which admits third order invariants and hence the PT is necessarily of first order. In order to get the correct space group symmetry change, four parameters should be set equal to zero and the spontaneous values of ti should be equal in magnitude, i. e., t;> f; = 5. Knowing the transformation properties of PT parameters, the free energy F can be constructed (throughout this paper a cubic coordinate system is used in which x, Y , = 1, 2, 3)
F = F o + & W ) ( t ? -I- tg) -I- i B ~ ( t ? + t3 +
+ s,itl + t i ) u,, + - tf) ( ~ x x - uYY) + + polarization and elastic energies.
Now the macroscopic properties of boracites near the PT can be studied in the usual way [lo]. The main results are : the spontaneous values of the polarization P: and strain components u:,, u:,, = u;, are pro- portional to f 2 . We emphasize that unlike in PF, in I F the spontaneous strains result rather from the coupling of uik with ti then from the normal electro- mechanical coupling with Pi. In other words, Pi and ug in I F may no longer be related by standard electro-
mechanical equations. Indeed Kobayashi et al. [ l l ] found experimentally that in Fe-I boracite diagonal strain components u:i are not related to P: through electrostriction. On the other hand in gadolinium molybdate (GMO) and in Fe-I boracite Pz and u&
are approximately related through normal piezoelec- tric coupling ; in Fe-I boracite because the coupling of u,, with ti is relatively small and in GMO because the coupling of P, with PT parameters is negligible [lo], [12]. This suggests that P: in Fe-I boracite arises due to direct coupling to ti while u;, results mainly through the normal piezoelectric coupling as a cr higher order secondary >) effect. (PS itself is a secondary effect and therefore it is in general smaller than in PF.) Just a reverse situation exists in GMO. This qualitatively explains why in Fe-I boracite P: is at least by an order of magnitude larger and u:, almost by an order of magnitude smaller than in GMO.
In the paraelectric phase all material constants of boracites should be temperature independant. Due to the coupling of P,, u,,,, uii with ta some of them change discontinuously at the PT temperature (first order PT) and then display a slight temperature dependence of the same type. For example, the clamped value of c,, is given by
&i3 refers to the paraelectric phase.
is positive and decreases with decreasing temperature due to the increase of t2. (The temperature course of
E~~ agrees qualitatively with experimental data 1131.) The following constants also should show temperature
P E
anomalies : cik, cik (i, k = 1, 2, 3, 6), &fj3ee and the piezoelectric constants a, ,, a,,, a 3 ~ .
The author thanks Dr. V. Janovec and Dr. J. Fousek of the Institute of Physics for many valuable discus- sions.
References [I] LYUBARSKI (G. Ya.), The Applications of Group
Theory in Physics, Pergamon Press, Oxford, 1961.
[2] INDENBOM (V. L.), SOV. Phys.-Crystallogr., 1960,5,106.
[3] AIZU (K.), Phys. Rev. B, 1970,2,754.
[4] LEVANYUK (A. P.) and SANNIKOV (D. G.), SOV. Phys.- JETP, 1969, 28, 134.
[5] HOLAKOVSK~ (J.), in preparation.
[6] ASCHER (E.), J. Phys. Soc. Japan, Supplement, 1970, 28, 7 (Proc. of the Second Inter. Meeting on Ferroelectricity, Kyoto, 1969).
[7] DVOGAK (V.), J. Phys. Soc. Japan, Supplement, 1970, 28, 252 (Proc. of the Second Inter. Meeting on Ferroelectricity, Kyoto 1969).
[8] DVOIIAK (V.), Phys. stat. sol. (b), 1971, 45, 147.
[9] DvokA~ (V.) and PETZELT (J.), Czech. J. Phys. B, 1971, 21, 1141.
[lo] DVOEAK (V.), Czech. J. Phys. B, 1971, 21, 1250.
[ l l ] KOBAYASHI (J.) and MIZUTANI (I.), Phys. stat. sol. (a), 1970, 2, K 89.
[12] DVOGK (V.), Phys. Stat. Sol. (b), 1971, 46, 763.
[13] ASCHER (E.), SCHMID (H.) and TAR (D.), Solid State Commu~., 1964, 2, 45.
