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Phase transitions in ferroelectric solid solutions of complex perovskites A. A. Bokov
a
a
Institute of Physics, Rostov State University, Rostov-onDon, 344104, USSR Available online: 08 Feb 2011
To cite this article: A. A. Bokov (1989): Phase transitions in ferroelectric solid solutions of complex perovskites, Ferroelectrics, 90:1, 155-163 To link to this article: http://dx.doi.org/10.1080/00150198908211285
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Ferroelectrics, 1989, Vol. 90, pp. 155-163 Reprints available directly from the publisher Photocopying permitted by license only
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PHASE TRANSITIONS IN FERROELECTRIC SOLID SOLUTIONS OF COMPLEX PEROVSKITES A. A. BOKOV Institute of Physics, Rostov State University, Rostov-on-Don, 344104, USSR
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(Received September IS, 1988) The phenomenological theory of ferro- and antiferroelectric solid solutions is proposed taking into account the possibility of compositional order variation. Complex perovskites AB;.5B:.503 where one of the cations is substituted are considered. The parameter of compositional long-range order in solid solutions is calculated in Bragg-Williams approximation. Formulae for the Curie temperature and permittivity composition dependences are deduced. The theory is in good agreement with the experimental data.
The possibility of the compositional disorder (i.e. the disorder of dissimilar atoms distribution in similar lattice positions) effect is usually not considered when analysing ferroelectric (FE) solid solution properties. This effect however may appear to be considerable. It has been e~tablishedl-~ that the degree of the compositional order in PbSc,,,Nb,,,O, and some other ternary oxides may vary over a wide range (owing to the order-disorder phase transition) depending on the temperature of samples annealing To. It is possible to control FE properties by changing the degree of the compositional order. It has been shown experimentally both for individual ternary and their solid solutions.4 This paper proposes phenomenological treatment of ferro- and antiferroelectric (AFE) solid solutions taking into account the possibility of the compositional order variation. The theory explains for example the existence of anomalies on the Curie point composition dependences provided isovalent substitution of one of the cations takes place (in solid solutions of binary perovskite oxides these dependences are usually linear). Let us consider the solid solutions of two ternary oxides xA’Bb.,B~,,O3-(1 - x) A’B;l,sB;;IS03 (I) or X A ’ B ; ) , , B ~ . ~ O~ -x)AB~.,B~,,O, (~ (11) on condition that the valences of B” and B”’-cations are identical but differ from the valence of B’; the valences of A’ and A-cations are also identical. The long-range ordering in perovskites is known to be observed only in the case of heterovalent cations., This is due to the fact that the main interaction leading to ordering is electrostatic and the energy of the interaction is proportional to the square of the ions’ charge difference (Aq)2.Let us then consider the long-range order in the arrangement of B” and B”’-ions (the same as A‘ and A”) to be absent and the degree of the compositional long-range order in solid solutions to be characterized by the parameter s = p?) - l where p:’) is the probability of B’-ions occurrence in the sites of one of the octahedral sublattices, each being filled only by isovalent B-ions in the completely ordered state. Let us write down the expression for solid solution free energy in the same form 155
A. A. BOKOV
156
as for the individual ternary oxide3 but introduce the composition dependence of expansion coefficients as is usually done6 when describing FE solid solutions:
F
=
Fo(x,s)
+ [ f o + fix + fcp + (91x)s21(p: + P%)
+ [go + g1x +
(i + il~)~*IP,Pb
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+ [ho + h,x + (x + x1x)s2](P,4 + P;) + [ro + r,x + (p + pIx)s2](P: + PZ)
(1)
It is kn0wn~9~ that the Curie point in the same crystal may be FE or AFE depending on the ratio of expansion coefficients values. Expansion coefficients in their turn depend on s . Ordered ~ compounds are known to be AFE, as a rule, and disordered ones-FE. So let us consider that the increase of s in solid solutions result in AFE state energy decrease as compared to FE. At some critical value s, ( x ) the Curie point may turn from FE into AFE (such change was found experimentally3 in PbIno,,Nbo503). Let 8, be the temperature boundary of paraelectric phase stability in the solid solution of composition x and order parameter s, s, and Ox, the critical values of s and 8 in the solid solution of composition x . In the case of paraelectric-ferroelectric phase transition ( s ( x ) < sxC) we get the f ~ l l o w i n g : ~
2[fo + fix + (9 + cp1x)s2I + go + g1x + (i + i1x)s2 = 2hl (T in the case of paraelectric-antiferroelectric transition (s(x) > sxc) 2[fo + fix + (cp + cp1x)s2I - go
-
g1x
-
(i + i1x)s'
=
2MT
- Ox,)
-
Ox,).
(2) (3)
Making certain transformations3 we get from Eq. (2) for s(x) < sxc:
The temperature boundary of paraelectric phase can be found by solving the system of two equations obtained from Eq. (4) as a result of substituting the critical value of s and x = 0; x = 1 taking into account Ox,, = 8,, + (8," - 8&:
By similar arguments it follows from Eq. (3) for the case s ( x ) > sxc:
PHASE TRANSITIONS OF COMPLEX SOLID SOLUTIONS
157
Composition dependences of s and 8 critical values can be calculated by solving the system of two equations ( 5 ) and (7) where s = sxc, BxS = €Ixc:
[(OOl ex,
=
-
011)H +
(810
-
%0)GI x + G - H
e00
G
-
H(G +
001)
.
(9)
Substituting s and x boundary values into the expression3for E in the paraelectric phase we get:
--+($A&+[(;-&)
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_1 Exs
1
€00
or in another form
where Ccw is the Curie-Weiss constant. The state when s = 1 is inaccessible. The crystal may also appear to melt at s > 0. In this case the values of ell, Oo0, OO1, Ole, E ~ q~ l ,, E ~ , can be found from the equations (9,(7), (10) with x = 0, x = 1 by means of substituting the values of s, O1s, OOS, qsrE~~ obtained experimentally in samples annealed at different temperatures. It is still not clear if the change of the Curie point from FE to AFE induced by s variation in a general rule for all perovskites. If this change is not observed in one or both end members then so, slc, OOc, el, may be considered phenomenological parameters providing realization of relationships ( 5 ) , (7-9). Let us perform statistical computation of s for the solid solution of type I1 within the scope of Bragg-Williams theory for alloys8 extended to pero~skites.~ Let N, (i = 1,2,3) be the number of ions of certain type (hereafter symbols with subscript 1 refer to B’-ions, 2 refers to B”-ions, 3 refers to B”’-ions);N = N, + N2 + N3 the general number of ions in octahedral sites; Nil), N!2) the number of i-type ions in the first and second octahedral sublattices respectively; p!’), the probability to find ions in these sublattices; v i k ,Nik ( k = 1,2,3) interaction energy of neighboring B-ions and the number of corresponding ion pairs; T, the temperature of compositonal order-disorder phase transition in solid solution with composition x. Let us then consider isovalent B”- and B“’-ions(at any value of s) to occupy two octahedral sublattices at random. In other words, p p ) = 0.5 (1 - s) (1 - x ) ; pi‘) = 0.5 (1 + s) (1 - x); p f ) = 0.5 (1 - s) x ; pi2) = 0.5 (1 + s)x. The configuration part of free energy F, (i.e. the energy connected with the compositional disordering) is determined in Bragg-Williams approximation:
F, = E
-
kTlnW
where E is the energy of the crystal with certain ion configuration, W
(11) =
the general
158
A. A. BOKOV
numbr of ions B‘, B”, B”’ distinguishable permutations in the first- and secondtype sites.
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The energy E was defined taking into account only the nearest neighbors’ interaction: (13) E = (Nllv11 + NZZVZZ + N33v33 + N12V12 + N13v13 + N23V23) The values of N, are found from the expressions N,, = 0.5ZN(p!1)pr+! + p!2)pp)) where Z is the coordination number (for the structure under consideration it equals 6). Substituting W and E from Eq. (12), (13) into Eq. (11) and making standard manipulations8 we get (NIv11
+ N21.322 + N3~33)+
-
N kT - (k(1- x)(l 2
-
2(1
+ (1
~
1
+ 3wp(1 ~
+ 21n2) + 21112 x)’ln (1 - x ) - [(l + s)ln(l + s) s)ln(l - s ) ] ( x 2 - x + l)}, --
1nN
-
- X)
2x2 lnx
-
-
(14)
where wik = 2vik - vii - Vkk. The temperature T, and the equation which enables to calculate the equilibrium value of s at Taxare found by applying the condition for minimum free energy dF,/ds = 0 (the sample sintered at To, and then quickly cooled will have the same value of s):
Substituting into Eq. (16) its values at x
=
0, x = 1 we get
where Q = 3wZ3/4k.As can be easily seen Q is an order-disorder transition ternperature in the individual oxide A B ~ , , B ~ , 0 As 3 . T,a(Aq)’ at a first approximation’ and B” and B”’-ions are isovalent we may roughly take Q = 0. When the sample having x = 0 does not achieve order-disorder transition at temperatures lower than the melting point, T, can be found from formula (15) provided the values of s at two annealing temperatures Ta0are known. T,, can be found similarly. The composition dependence of T, for ternary perovskite solid solutions has a maximum according to Eq. (117). It may result in emergence of considerable anomalies on the composition dependences of s and of 8 and other FE parameters
1740
1480
Tin (K)
2700
2250
T,, (K)
310
340
390 265
001
(K)
000
(K)
o,,,
220
650
(K)
01I
370
590
(K)
Of,
295
270
(K)
180
480
8,' (K)
0.95
0.95
SIh
0.6
0.7
SI
