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The sequence of structural phase transitions in KTa1−xNbxO3
PHYSICA Physica B 222 (1996) 182-190
ELSEVIER
The sequence of structural phase transitions in KTal-xNbxO3 M , D . G l i n c h u k a'*, L. J a s t r a b i k b, V.A. S t e p h a n o v i c h c aInstitute for Materials Science, Ukrainian ALAS, Krgiganovskogo str., 3, 252180, Kiev, Ukraine bInstitute of Physics, Academy of Sciences of Czech Republic, Na Slovance 2, 18040 Prague 8, Czech Republic Institute of Semiconductor Physics, Ukrainian NAS, Prospect Nauki, 45, 252028, Kiev, Ukraine
Received 18 April 1995;revised 6 November 1995
Abstract The series of structural phase transitions in KTal-xNbxOs is described by a modified Landau free energy functional. The modification is that the phase transition temperature Tc depends on the Nb concentration. The dependence Tc(x) is calculated self-consistently by averaging over spatial and thermal fluctuations in Nb subsystem. Good coincidence between theory and experiment is manifested.
1. Introduction Recently it was experimentally discovered [1-3] that KTal_xNbxO3 (KTN) at x > 0.02 undergoes three structural phase transitions within a temperature range, where spontaneous polarization exists. Upon lowering the temperature, the crystal becomes tetragonal, orthorhombic and rhombohedral from paraelectric cubic phase, respectively. The temperatures of such structural phase transitions of first order differ from each other by 2-3 K [1-3]. Although the presence of structural phase transitions within the polar phase is well known for conventional ferroelectrics like BaTiO3 (see, e.g. Ref. [4]), they have many peculiarities in K T N due to effects of concentrational disorder. For example, at x < 0.01 long-range order in K T N is impossible, so that the system is in dipole glass state (see Ref. [3]). At 0.01 < x <0.02 K T N has ferroelectric long-range order in rhombohedral phase only, i.e. structural phase transitions do not take place [4]. The latter effects cannot be explained in the framework of ordinary Landau theory of phase *Corresponding author.
transitions which was used for their description in BaTiO3 (see Ref. [4]). In the present paper, we suggest a mixed model for the description of the peculiarities of the structural phase transitions in KTN. This model can be regarded as a generalization of Landau's phase transition theory to the case of disordered dielectrics. The essence of our approach is that we use for this purpose a Landau free energy functional with a multicomponent order parameter, where, however, the ferroelectric phase transition temperature Tc depends on the Nb concentration. Tc(x) is calculated by means of a statistical method (see Refs. [5, 6]), which permits to take into account both spatial disorder in the Nb subsystem of KTaO3 and thermal fluctuations of each Nb off-center impurity. This way we derive the temperatures of the tetragonal-orthorhombic and orthorhombic-rhombohedral phase transitions, where the free energies of both phases involved are equal to each other.
2. Theory At high temperatures (cubic phase), the KTaO3 crystal has m3m symmetry, so that free energy
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M.D. Glinchuk et al./Physica B 222 (1996) 182-190
density of KTN has the form:
F =Fo + Z-~(x)(p~ + p2 + p~) + B~(P~ + P~ + P~) 2 2 2 2 2 2 + B2(P~Pr + PxPz + PyP~)
+ C,(p6 + e6r + p6) 4 2 4 2 + C:(Px(Pr + p2) + Pr(Px + p2) 4 2 + Pz(Px + P~)) 2 2 2 + C3P~PrP~,
B1, B2 < O,
(1)
where Fo is free energy in paraelectric phase (i.e. at P = 0); the coefficients B, and C, should be taken from the condition of best fit to experiment; X(x) is the dielectric susceptibility of the system under consideration in the high-temperature phase. It can be shown that, for our purposes, it is suffÉcient to put B. and C, to be independent of concentration and temperature; only Z- ~(x) has to be calculated. The calculation of Z - Xz~ for the system with Ising-like dipole impurities [7] shows that for all x it can be well represented by Curie-Weiss-like law
X- l(x) = A(x)(T - T~(x)) = a(x),
p6(i = x,y,z). The equilibrium value of the spontaneous polarization can be obtained by free energy [Eq. (1)] minimization. To do so, we need to satisfy both the conditions of extremum
~P,
-
0,
(2)
and of minimum. The latter condition reads that for the points Po, where condition (2) is satisfied, the second differential of free energy should be a positively defined quadratic form [8] d2F --
~ i,k=x,y,z
O2F O'------~k OP, dP~ dPk > O.
/
/
/
/
/
~
Fig. 1. Schematic plot of the spontaneous polarization vector directions in the tetragonal (T), orthorhombic (R) and rhombohedral (RE) phases of KTN.
Extremum condition (2) gives the following system of equations for P
~P~ + 2B~P~ + B2P~(P~ + P~) + 3C1P~
(la)
where A(x) and T¢(x) have to be calculated. Note that with the exception of the concentrational dependences of A and T¢, Eq. (la) has the ordinary form known from Landau's phase transition theory, Since we use Eq. (1) for the description of a first order concentration dependent phase transition in KTN, it includes all invariants up to order
~F
f
(3)
3 2 + p2) + C:Px(P~ + P~) + 2C2P~(Pr 2 2 + CsP~P>,P~ = O,
~py + 2Bip3 + B2py(p2 + p2) + 3CIp5 3 : + P~) + C2Py(P~ + p4) + 2CzPr(Px 2 2 + C3PrPxPz = O,
c~Pz + 2B1Paz + B2Pz(P 2 + p2) + 3C,pSz 3 2 + 2CEPz(Px + p2) + C2Pz(P~ + P~)
+ C3Pzp2p 2 = 0,
~ = A(T - To(x)).
(4)
It is easy to show that the system of equations (4) has four families of solutions, corresponding to zero polarization (cubic symmetry); polarization directed along the fourth order axis (tetragonal symmetry); along the second and the third order axis (orthorhombic and rhombohedral symmetries, respectively) (Fig. 1). To be more specific, for all
M.D. Glinchuk et al. / Physica B 222 (1996) 182-190
184
aforementioned symmetries we have
Pz=PT; FT = otP2T+ B,P~ + C l P 6 , (5) Px=Py=PR, n~=0; FR = 2o~e2 + (2B1 + B2)P~ + 2(C~ + C2)p6; (5a) Px= P,= P,= PRE; FRE = 3ap2E + 3(B~ + B2)p4E Px = Pr = 0 ,
+ (3C1 + 6C2
(5b)
+ C3)p6E •
The equilibrium polarization values for tetragonal, orthorombic and rhombohedral phases can be obtained from Eqs. (5), (5a) and (5b) with respect to Eqs. (2) and (3). For all these symmetries the polarization components Pi(i = T, R, RE) have the form
P' = + ~]
2C, X~ +
J 1 4B~c, "
where rc is pure KTaOa correlation radius. Note that the second term in the function fl is added for the boundary conditions for the shorter sample to be correct [9, 10]. As it was shown in Ref. [11], this term is completely defined by the anharmonicity coefficient of the host lattice. In the same reference the expressions for fl and f2 with respect to anharmonicity were calculated. Here we do not repeat them because they have a cumbersome form. The hamiltonian (9) can be rewritten identically introducing the random local field E~, acting on each dipole from the remaining impurities.
E, a = ~ ~'tJ(rij)l~. j~
(10)
Let us now introduce the distribution function of the random fields
(6)
f (E) = ( 6(E -- El)),
(11)
Here
Bx = 2BI,
CT = 3C1,
BR = 2B1 + B2,
CR = 3(C1 + C2),
BRE=B1
CRE =
+B2,
3C1 + 6C2 +
(7) C 3•
The next step is the calculation of To(x). To do so, let us write the hamiltonian of K T N in the form (see Refs. [9, 10])
where the bar denotes averaging over spatial positions of Nb impurities and angular brackets the thermal average over their eight discrete orientations. Spatial averaging can be fulfilled with the help of the &function integral representation
f(E)=
1 f°~ f ~ --o0
--
X
(ri, j ) l i l j ,
~,fl = x , y , z ,
--o0
f°° exp(iEp) --o0
(8)
(12)
x(expl-i~p~Ei~lld3p. where li is the unit vector directed along the offcentre Nb dipole moment d* in the point ri(d* = d*li), d* = ½y deo is the Nb effective dipole moment (see also Ref. [10]), 7 is the Lorentz factor, eo is the static dielectric permittivity of the pure crystal, 0~,fl = x, y, z and rii = rj - ri. The sum in Eq. (8) is running over all KTaO3 sites occupied by Nb ions. In our calculation of T~ we shall take :;ff~(rij) in the form [9] o'~fl(rij) =
d*2 [ fl (rij)6~ + (3n~np -- 6~p)f 2 ( r i j ) ] , Bo
n = -rij
rij'
f2(x) = __ xv
f l (x) = 2 exp(--x/re) -~ 4x
3xr 2
1 - exp(--x/r¢)
3V '
1 +-- + re
(9)
Further spatial averaging gives
~Ilfv(expl-i~p~E~lldVl =II+lfv(exp(-i~p~E~)-lldVl
~ N. (13)
Introducing the impurity concentration n = N / V and passing to thermodynamic limit N--.0%
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M.D. Glinchuk et al./Physica B 222 (1996) 182-190
Here we used the fact that X~a = ~ff#~(see Eq. (9)). A~er convertion of hyperbolic functions with imaginary arguments into trigonometric ones with respect to Eq. (15) we have finally from Eq. (16)
V ~ ~, n = const., we obtain (exp I - - i ~ P~E'~I)
/exp I - i =~ Jd=#Pflal - 1 )
=explnfv(eXp(-i~p~E~)-l)dVl.(14)
= cos Q, cos Qr cos Qz - 1 Here we convert the sum over the lattice into an integration. The approximate equality sign in Eq. (13) means that to obtain Eq. (14) from Eq. (12) we neglected higher order (more than pair) correlations of Nb impurities. As it was shown in Refs. [5, 6] this neglect does not play a crucial role for small enough impurity concentrations. Averaging of Eq. (14) over thermal fluctuations of Nb impurities is performed in two steps. First we shall calculate the auxiliary thermal averages (l~) and (l 2) with the one-particle hamiltonian ~ = 1E, I being oriented along the eight directions of [1 1 1] type, E is the argument of the function (12), (l.,r,~)
=
--
( l x , y , z ) = 1, 2
tanh(fE~.r.z); f=(kBT)
(15)
-~
Note that, in spite of the fact that we start from microscopic Nb orientations along the directions of [1 1 1] type, the macroscopic spontaneous polarization orientations in our model are determined by free energy (1) minimization, i.e. the macroscopic behaviour is governed by coefficients B, and C~. We have now from Eq. (14) /exp I - i ~a
~,#P~,I#I) =
cosh (flEx - iQ~) cosh (fE r - iQr ) cosh (fEz - iQ=) cosh (fE~) cosh (tiEr) cosh (fEz)
Oz = Yd~zpx + YCzp~ + YCzzpz.
-
-
(lx) (lz) sin Qx sin Q= cos Qy (ly) (l=) sin Q= sin Qy cos Qx
+ i [ (lx) sin Qx cos Qr cos {2= + (1,.) sin Qycos Q~cos Q~ + (l~) sin Qz cos Qy cos Qx - (lx)(ly)(t~)sinQxsinQysinQ~].
(17) The next step is to obtain of equations for components of average system dipole moment L, = (l,). The latter quantities can be regarded as spontaneous polarization components. Knowledge of them actually permits to determine the equilibrium characteristics of the system like dielectric permittivity, specific heat, etc. (see Refs. [5, 6]). In our approach, however, we shall use the equations for L, to calculate T¢ as the temperature, where L~ # 0 appears. To obtain the equations for L,, we shall use a self-consistent procedure, i.e. express distribution function (11) through itself. In our opinion, the easiest way to do so is to write the equations for L~ in the form L x. r. z =
(lx. r., ) f (E, L) dEx dE r dE=, -
oo
-
ct3
oo
(18)
Q.~ = x ~ p x + YCypy + Ydx=pz; gy = XxrPx + . ~ y P r + ~ff, zPz;
(Ix) (ly) sin Q,, sin Qy cos Qz
-
(16)
where ( l , ) are determined by Eq. (15). f ( E , L ) corresponds to function (11), calculated with respect to Eqs. (14) and (17) but replacing (1,) by L, in Eq. (17). This procedure gives the following final ex-
186
M.D. Glinchuk et al./Physica B 222 H996) 182-190
pression for the distribution function
f(E,t)= 1 fj f°~ f~ exp(iEp)exp[F~(p) oo
--
o~3
--
o0
+ iFz (p)] dpxdprdp~, F1 (p)
n fv [cos Q~ cos Qr cos Qz -
-
Fz(P)
1 fo DpEo(p)exp(Fl(p)) dp, tic = (k,Tc) -1 sinh (np/2fl~)
1= ~
(22)
1
Note that in other words we had shown that the system of Eqs. (18) and (19) has solutions corresponding to tetragonal (Lx = L; L y - - L z = 0, etc), orthorhombic and rhombohedral phases which are determined, however, by a single equation for L [Eq. (19)] and Tc [Eq. (22)]. That is why Eq. (22) with equal probability determines the temperatures of transitions to tetragonal, orthorhombic and rhombohedral phases. It gives the equation for dependence
- L~Ly sin Q~ sin Qycos Qz -
The equation for the phase transition temperature Tc can be obtained from (20) by inserting (21) in the limit L ~ 0. This yields
L~L~ sin Q~sin Qzcos Qr
LyLz sin Qz sin Qr cos Q~] d3r,
n fv [L~ sin Q~cos Qycos Q~ + Lr sin Qy cos Q~ cos Q~
T~ = f (nr3),
+ Lz sin Qz cos Qx cos Qr
(22a)
TcMF -
LxLrLz sin Qx sin Qy sin Q~] d3r, (19)
which determines the temperature and concentrational dependences of the average macroscopic dipole moment of KTN. The Eqs. (18) and (19) can be simplified if we take into account the fact that (Ix) in Eq. (15) depends on Ex only (similarly (ly) and (l~) depend on Er and E,, respectively). So, we can obtain (see Appendix for details) 1 fo ~ sin (F2 (p)) exp (F1 (p)) dp.
L = -fl
(20)
sinh(np/2fl)
The aforementioned fact permits also to simplify the expressions for FI(p) and F2(p). We have
where nr3 is a dimensionless parameter determined by the Nb concentration, TcMF is the phase transition temperature in the mean field approximation. It is determined from the equation (see Refs. [9, 10])
4nnd .2 kBTcMF -- 3e0(T~MF)"
(22b)
As T¢ ~ 0 one can obtain the equation for the Nb critical concentration (nrac)¢rfrom Eq. (22). Let us remind that long-range order occurs in KTN at T = 0 if n > net; in the opposite case, the dipole glass state is realized. The equation for (nrca)~rreads
1 = ~2fo° Eo(p)exp(Fl(p)) dp.
(23)
Fa,2(Px) = FL2(pr) = FL2(p~) = FL2(p), F2(p) = LpEo(p), Fx(p) =
(21)
g o
The numerical solution of this equation with respect to Eq. (9) gives (nr~)~r -~ 0.0184.
(24)
n Jv [cos (Jd~zp) cos (~ffyzP)cos (o~ffzzp)-- 1] d3r,
Eo (p) = ~n f v sin ( ~ 0 ) cos (o~y~p)cos (X~zO)d3r.
The dependence (22a) is shown in the Fig. 2. It will be used in the next section for comparison with experiments on the structural phase transitions in KTN.
M.D. Glinchuk et al./Physica B 222 (1996) 182-190
187
where T is the temperature in Kelvin, G = 6000-8500 cm-2. For our calculations we choose G = 6500 cm- 2. Thus we can rewrite Eq. (22a) in the form
-['c/TcM=
To(x) =
1
10
(nr~)-I
TcMF(X)f(x~(re(~raC))3) ,
(26)
where r~(T~) is determined by Eq. (25). ~ ~- 76.21 is introduced to match the theoretical value (nr3)~r [Eq. (24)] with the experimental one (see Ref. [3]) Xcr ~-- 0.01. The discussion of the reasons why the theoretical value of critical concentration is smaller than the experimental one (this corresponds to = 1 in Eq. (26)) is found in Refs. [5, 6]. The explicit form of equation for T~ME(X)can be obtained from Eq. (22b) by inserting Eqs. (25) and (25a). It reads
Dx
Fig. 2. Dependence of Tc/TcMF on nr 3.
TcMv = 27.1 coth (27.1 / TeMF)
--
12.9'
(27)
where
3. Comparison between theory and experiment; discussion The first step in an adjustment of our theory to the experiment is to obtain the dependence To(x) from Eq. (22a). For the conversion of the parameter nr~ to the atomic concentration x we use the relation
nr3 = x (rc~ 3 ,,a/
(25) '
where a ~- 4 A and r~ are the lattice constants of pure KTaO3 and the correlation radius (see above), respectively; the latter quantity is temperature dependent. We shall calculate the temperature dependence of r~ using the Barrett formula for the dielectric permittivity (or soft mode frequency e)s~) of pure KTaO3 (see Ref. [12]).
t; 0 ~- (D2M '
COs2M--- 26.618 [27.1 coth (~--~) - 12.9] (cm-Z), (25a)
D -~ 5.33 x 108 (Tde)2 ka
(28)
o
Here d (in eA) is the Nb off-center impurity dipole moment; e is the charge of an electron. The constant D can be determined from a well-established experimental point T~(x) at sufficiently large x (x ~_ 0.1), where the mean-field approximation is valid. Such a fit depends on merely one parameter, yd. Since the off-center displacement of Nb is quite well known (see Refs. [13, 14]), this procedure can be considered as that for the experimental determination of, the Lorentz factor 7 at an Nb site. This possibility seems important, because theoretical calculations of 7 (actually 7 = 7(r)) require microscopic models of crystal lattice including consideration of the electronic structure of atoms and ions, interatomic forces, etc (for some details see Refs. [13,14]). It is clear that a model of such kind requires tedious numerical calculations. Obtaining 7 values in such a way depends strongly on the model, which is used for the calculation, so that there is a large difference in their theoretical values. So, in our opinion it is important to start from an independent experimental determination of 7-
M.D.Glinchuketal./PhysicaB222(1996)182-190
188
impurity concentration and other external conditions, which has the smallest free energy. To fulfill this program and to reduce the number of parameters, we shall rewrite the Eqs. (5)-(7) into dimensionless variables. This yields
100.00 @ L.. -,.i-o
U 80.00 L_
.S Z
Q.
E ~)
1
cO •-
x6 '
x 2 = (1 - 0) 1/2 + 1;
60.00
~DR
iiI/~ iiII/~'
40.00
C 0
.~-
20xzR + (2 + r/,)xR4 + 2(1 + r/¢l)x6 ,
2+r/B ( ~1/+ ~c2 - 2(1 + r/c, )
1
40(1 + t/m)'] ; (-2+t/B--~)
L,,.
-@ 20.00 ol 0 _C
o_ 0.00
/ I/
// f
IIl~'h
0.00
iiii1~
zoo
x2 ~ = iitll
~lll
Ilrllllllblllllllll
4oo 6.00 Nb c o n t e n t
Ill~
8.oo
Illll
+ 3(I + r/.)x&
+ 6r/o, +
3(1 + r/.) 3 + 6q¢i + r/¢2
I
o.oo
Fig. 3. Temperatures of transitions to tetragonal (T), orthorhombic (R) and rhombohedral (RE) phases in K T N versus N b content. Solid line: our theory; *: experimental points from Ref. [-3]. Dashed line corresponds to the dependence TCr~v(X).
x ( l + \ / 1 0 ( 3 + 6 r / c~-1--~£-~2l + r / c 2 ) ' ~ - .//, where
Bz r/B =-~1, 0 =
From the experimental value TcMF(X = 0.09) = 95 K [2] we get D = 80 690o K. Hence, dy ~ 0.3 so that o 7 " 3 at d - ~ 0 . 1 e A and 7-~0.6 at d ~ 0.5 cA. The dependence TCMF(X), calculated by solving Eq. (27) with respect to the above obtained D value, is shown in Fig. 3. Now we can calculate Tdx) from Eq. (26). We consider this quantity from the point of view of describing the phase transitions of KTN as temperature of transition from the cubic paraelectric phase, where P = 0, to the first phase, where P ¢ 0, i.e. to tetragonal one. Such supposition is in coincidence with the physical meaning of the equation for T¢ [Eq. (22)]. Let us determine theoretically the temperatures of the structural phase transitions tetragonal-orthorhombic and orthorhombic-rhombohedral. As is usual in Landau phenomenological theory of phase transitions, we shall define these temperatures as those for which free energies of corresponding phases are equal to each other. Note that that phase is stable at a given temperature,
(29)
qgi-
C2,s r/c,.: = C1 '
3~C1
= Oo(x)(T - To(x)),
= 3C1B,
Bal Fi.
Since expression (la) for Z -1 is approximate, dependence Oo(x) cannot be calculated exactly. An approximate form of Oo(x) can be extracted from calculations of Zu "~ OLi/OEj (E is an external electric field), see Ref. [7] for details. Such extraction, made from Z=, calculated in Ref. [7] for a model system with two-orientable dipoles, gives Oo(x) as an expansion into a power series of x. In our case of eight-orientable Nb dipoles it can be shown that Oo(X) will be qualitatively the same. For our purposes it is sufficient to keep terms up to order x 2. The coefficients of this expansion can be extracted for our case from the conditions of the best fit to the experiment with respect to conversion nr~ -+ x (see Eq. (25)). So, we have finally for Oo(X)
Oo(x) ~ 556x 2 - 96.7x + 5.6.
(30a)
This expression together with values r/B ~ 1.65;
qCl ~ 3.5;
r/c2 ~ - 1
(30b)
M.D. Glinchuk et al./Physica B 222 (1996) 182-190
gives a complete determination of the desired free energy. It can be shown that conditions (30a) and (30b) can be satisfied if and only if we put TcR ,~ 0.96T¢;
T¢RE ~ 0.4T¢,
(30c)
where T~R and ToRE are transition temperatures to orthorhombic and rhombohedral phases, respectively, in the case, when the other (i.e. rhombohedral and orthorhombic, respectively) phase would not exist. It means that one has to substitute the values of T~R and TCREto ~0Rand ORE, respectively, instead of T¢ (see Eq. (26)). Introduction of T~R and ToRE have important physical meaning: for TcR < T < T~REwe have long-range tetragonal order with nuclei of orthorhombic phase; these nuclei grow and give rise to the orthorhombic phase at T = TcR; the situation for the rhombohedral phase is similar. It means, that condition (30c), which was obtained from mathematical reasons, coincides with the physical picture of the first-order structural phase transitions in KTN. The calculation of the concentrational dependences of the temperatures of structural phase transitions in K T N can be done as follows. The solution of Eq. (26) with respect to Eq. (27) gives Tc values for each x point. With the help of these values for the same x points we can find the temperatures, where OpT= ~PR and 9R = ~gRE. They are indeed the desired temperatures of structural phase transitions. The theoretical curves Ta(x) are shown in Fig. 3 together with experimental points, taken from Ref. [3]. Good coincidence between theory and experiment is realized. It is also seen that theory differs from experiment for X¢r < X < 0.02. Whereas the experiment shows absence of the structural phase transition sequence in this region (one point instead of three at x > 0.02), theory predicts its presence up to x = x , . The first reason for this discrepancy is the coexistence between the ferroelectric phase and the dipole glass state (see Ref. [15, 16] for details). The second one is the difference in the Nb dipoles reorientation rates (between their eight possible directions) at low and high temperatures (see Ref. [10, 13, 14]). The latter reason results in only one low temperature transition from the paraelectric cubic phase to the rhombohedral ferroelectric one, since the Nb dipoles have to be regarded as non-
189
reorientable, i.e. static. Since we did not account for these reasons, our consideration is valid only at x ~> 0.02, where the structural phase transitions occur.
Note that, for a proper distinction between the dipole glass state and the ferroelectric phase, relaxation measurements are necessary. That is because dielectric relaxation in dipole glasses has long-time character, while in ferroelectrics it is of the conventional Debye type (see, e.g. Refs. [14, 16]). Note also, that the suggested calculation of the KTN free energy in the tetragonal, orthorhombic and rhombohedral phases can be applied for the description of its equilibrium physical properties. It can be used in particular for the description of nonlinear effects in the polarization P(E) and dielectric susceptibility x(E) especially near T --~ T c, where nonlinear effects are essential.
Acknowledgements The research described in this publication was made possible in part by Grants ISSEP SP0042015 and No. U4B200 (M.D.G.) and No. U68000 (V.A.S.) from the International Science Foundation.
Appendix Let us begin with the equation for L [Eq. (18)]. We have for Lx Lx - (27r)3
d3E
tanh (flEx) exp ripE)
x exp [Fl(p) + iF2(p)] d3p.
(A.1)
Integration over d3E in (A.1) can be done as follows
I(p) = ~
tanh(flEx) --or3
--cO
--oC
x exp [i(pxE~ + prEy + pzE~)] dExdEydEz
IA.2)
M.D. Glinchuket al./PhysicaB 222(1996)182-190
190
=(21rt)6(Py)6(Pz)f~o tanh(flEx)sin(pxEx)dEx = For
I(p)
(2re)
~
16(Py)~(pz)"
c a l c u l a t i o n we use the i d e n t i t y exp (ikx) dk = ~(x)
(A.4)
--oO
a n d Ref. [16]. S u b s t i t u t i o n of (A.2) to (A.1) gives Eq. (20) for L - Lx. T h e e q u a t i o n s for Ly,z c a n be d e r i v e d s i m i l a r l y to t h a t for Lx. It c a n be s h o w n t h a t Lx = Ly = Lz.
References [1] D. Sommer, D. Friese, W. Kleemann and D. Rytz, Ferroelectrics 124 (1991) 231. [-2] M.D. Fontana, E. Bouziane and G.E. Kugel, J. Phys.: Condens. Matter 2 (1990) 8681. [3] W. Kleemann, Int. J. Modern Phys. B 7 (1993) 2469. [-4] A.S. Sonin and B.A. Strukov, Vvedenie v segnetoelektrichestvo (Introduction to Ferroelectricity) (Vysshaya Shkola, Moscow, 1970).
[5] V.A. Stephanovich, Free energy functional of disordered dielectrics with dipole impurities, Institute for Theoretical Physics Ukrainian NAS, Kiev, 1994, Preprint ITP-94-62E. [6] V.A. Stephanovich, Ferroelectrics, to be published. [7] B.E. Vugmeister and V.A. Stephanovich, Solid State Commun. 63 (1987) 323. [8] G. Korn and T. Korn, Handbook of Mathematical Functions for Scientists and Engineers (Wiley, New York, 1968). I-9] B.E. Vugmeister and M.D. Glinchuk, Sov. Phys. JETP 52 (1980) 482. [10] B.E. Vugmeister and M.D. Glinchuk, Sov. Phys. Uspekhi 28 (1985) 589. [11] V.A. Stephanovich, Phys. Rev. B 47 (1993) 9084. [12] M.E. Lines and A.M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, Oxford, 1977). [13] B.E. Vugmeister and M.D. Glinchuk, Rev. Mod. Phys. 82 (1990) 993. [14] U.T. H6chli, K. Knorr and A. Loidl, Adv. Phys. 39 (1990) 405. [15] M.D. Glinchuk and V.A. Stephanovich, J. Phys.: Condens. Matter 6 (1994) 6317. [16] M.D. Glinchuk and V.A. Stephanovich, Ferroelectrics 169 (1995) 281. [17] I.S. Gradstein and I.M. Ryzhik, Tables of Integrals, Sums, Series and Multiples (Wiley, New York, 1965).
ELSEVIER
The sequence of structural phase transitions in KTal-xNbxO3 M , D . G l i n c h u k a'*, L. J a s t r a b i k b, V.A. S t e p h a n o v i c h c aInstitute for Materials Science, Ukrainian ALAS, Krgiganovskogo str., 3, 252180, Kiev, Ukraine bInstitute of Physics, Academy of Sciences of Czech Republic, Na Slovance 2, 18040 Prague 8, Czech Republic Institute of Semiconductor Physics, Ukrainian NAS, Prospect Nauki, 45, 252028, Kiev, Ukraine
Received 18 April 1995;revised 6 November 1995
Abstract The series of structural phase transitions in KTal-xNbxOs is described by a modified Landau free energy functional. The modification is that the phase transition temperature Tc depends on the Nb concentration. The dependence Tc(x) is calculated self-consistently by averaging over spatial and thermal fluctuations in Nb subsystem. Good coincidence between theory and experiment is manifested.
1. Introduction Recently it was experimentally discovered [1-3] that KTal_xNbxO3 (KTN) at x > 0.02 undergoes three structural phase transitions within a temperature range, where spontaneous polarization exists. Upon lowering the temperature, the crystal becomes tetragonal, orthorhombic and rhombohedral from paraelectric cubic phase, respectively. The temperatures of such structural phase transitions of first order differ from each other by 2-3 K [1-3]. Although the presence of structural phase transitions within the polar phase is well known for conventional ferroelectrics like BaTiO3 (see, e.g. Ref. [4]), they have many peculiarities in K T N due to effects of concentrational disorder. For example, at x < 0.01 long-range order in K T N is impossible, so that the system is in dipole glass state (see Ref. [3]). At 0.01 < x <0.02 K T N has ferroelectric long-range order in rhombohedral phase only, i.e. structural phase transitions do not take place [4]. The latter effects cannot be explained in the framework of ordinary Landau theory of phase *Corresponding author.
transitions which was used for their description in BaTiO3 (see Ref. [4]). In the present paper, we suggest a mixed model for the description of the peculiarities of the structural phase transitions in KTN. This model can be regarded as a generalization of Landau's phase transition theory to the case of disordered dielectrics. The essence of our approach is that we use for this purpose a Landau free energy functional with a multicomponent order parameter, where, however, the ferroelectric phase transition temperature Tc depends on the Nb concentration. Tc(x) is calculated by means of a statistical method (see Refs. [5, 6]), which permits to take into account both spatial disorder in the Nb subsystem of KTaO3 and thermal fluctuations of each Nb off-center impurity. This way we derive the temperatures of the tetragonal-orthorhombic and orthorhombic-rhombohedral phase transitions, where the free energies of both phases involved are equal to each other.
2. Theory At high temperatures (cubic phase), the KTaO3 crystal has m3m symmetry, so that free energy
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183
M.D. Glinchuk et al./Physica B 222 (1996) 182-190
density of KTN has the form:
F =Fo + Z-~(x)(p~ + p2 + p~) + B~(P~ + P~ + P~) 2 2 2 2 2 2 + B2(P~Pr + PxPz + PyP~)
+ C,(p6 + e6r + p6) 4 2 4 2 + C:(Px(Pr + p2) + Pr(Px + p2) 4 2 + Pz(Px + P~)) 2 2 2 + C3P~PrP~,
B1, B2 < O,
(1)
where Fo is free energy in paraelectric phase (i.e. at P = 0); the coefficients B, and C, should be taken from the condition of best fit to experiment; X(x) is the dielectric susceptibility of the system under consideration in the high-temperature phase. It can be shown that, for our purposes, it is suffÉcient to put B. and C, to be independent of concentration and temperature; only Z- ~(x) has to be calculated. The calculation of Z - Xz~ for the system with Ising-like dipole impurities [7] shows that for all x it can be well represented by Curie-Weiss-like law
X- l(x) = A(x)(T - T~(x)) = a(x),
p6(i = x,y,z). The equilibrium value of the spontaneous polarization can be obtained by free energy [Eq. (1)] minimization. To do so, we need to satisfy both the conditions of extremum
~P,
-
0,
(2)
and of minimum. The latter condition reads that for the points Po, where condition (2) is satisfied, the second differential of free energy should be a positively defined quadratic form [8] d2F --
~ i,k=x,y,z
O2F O'------~k OP, dP~ dPk > O.
/
/
/
/
/
~
Fig. 1. Schematic plot of the spontaneous polarization vector directions in the tetragonal (T), orthorhombic (R) and rhombohedral (RE) phases of KTN.
Extremum condition (2) gives the following system of equations for P
~P~ + 2B~P~ + B2P~(P~ + P~) + 3C1P~
(la)
where A(x) and T¢(x) have to be calculated. Note that with the exception of the concentrational dependences of A and T¢, Eq. (la) has the ordinary form known from Landau's phase transition theory, Since we use Eq. (1) for the description of a first order concentration dependent phase transition in KTN, it includes all invariants up to order
~F
f
(3)
3 2 + p2) + C:Px(P~ + P~) + 2C2P~(Pr 2 2 + CsP~P>,P~ = O,
~py + 2Bip3 + B2py(p2 + p2) + 3CIp5 3 : + P~) + C2Py(P~ + p4) + 2CzPr(Px 2 2 + C3PrPxPz = O,
c~Pz + 2B1Paz + B2Pz(P 2 + p2) + 3C,pSz 3 2 + 2CEPz(Px + p2) + C2Pz(P~ + P~)
+ C3Pzp2p 2 = 0,
~ = A(T - To(x)).
(4)
It is easy to show that the system of equations (4) has four families of solutions, corresponding to zero polarization (cubic symmetry); polarization directed along the fourth order axis (tetragonal symmetry); along the second and the third order axis (orthorhombic and rhombohedral symmetries, respectively) (Fig. 1). To be more specific, for all
M.D. Glinchuk et al. / Physica B 222 (1996) 182-190
184
aforementioned symmetries we have
Pz=PT; FT = otP2T+ B,P~ + C l P 6 , (5) Px=Py=PR, n~=0; FR = 2o~e2 + (2B1 + B2)P~ + 2(C~ + C2)p6; (5a) Px= P,= P,= PRE; FRE = 3ap2E + 3(B~ + B2)p4E Px = Pr = 0 ,
+ (3C1 + 6C2
(5b)
+ C3)p6E •
The equilibrium polarization values for tetragonal, orthorombic and rhombohedral phases can be obtained from Eqs. (5), (5a) and (5b) with respect to Eqs. (2) and (3). For all these symmetries the polarization components Pi(i = T, R, RE) have the form
P' = + ~]
2C, X~ +
J 1 4B~c, "
where rc is pure KTaOa correlation radius. Note that the second term in the function fl is added for the boundary conditions for the shorter sample to be correct [9, 10]. As it was shown in Ref. [11], this term is completely defined by the anharmonicity coefficient of the host lattice. In the same reference the expressions for fl and f2 with respect to anharmonicity were calculated. Here we do not repeat them because they have a cumbersome form. The hamiltonian (9) can be rewritten identically introducing the random local field E~, acting on each dipole from the remaining impurities.
E, a = ~ ~'tJ(rij)l~. j~
(10)
Let us now introduce the distribution function of the random fields
(6)
f (E) = ( 6(E -- El)),
(11)
Here
Bx = 2BI,
CT = 3C1,
BR = 2B1 + B2,
CR = 3(C1 + C2),
BRE=B1
CRE =
+B2,
3C1 + 6C2 +
(7) C 3•
The next step is the calculation of To(x). To do so, let us write the hamiltonian of K T N in the form (see Refs. [9, 10])
where the bar denotes averaging over spatial positions of Nb impurities and angular brackets the thermal average over their eight discrete orientations. Spatial averaging can be fulfilled with the help of the &function integral representation
f(E)=
1 f°~ f ~ --o0
--
X
(ri, j ) l i l j ,
~,fl = x , y , z ,
--o0
f°° exp(iEp) --o0
(8)
(12)
x(expl-i~p~Ei~lld3p. where li is the unit vector directed along the offcentre Nb dipole moment d* in the point ri(d* = d*li), d* = ½y deo is the Nb effective dipole moment (see also Ref. [10]), 7 is the Lorentz factor, eo is the static dielectric permittivity of the pure crystal, 0~,fl = x, y, z and rii = rj - ri. The sum in Eq. (8) is running over all KTaO3 sites occupied by Nb ions. In our calculation of T~ we shall take :;ff~(rij) in the form [9] o'~fl(rij) =
d*2 [ fl (rij)6~ + (3n~np -- 6~p)f 2 ( r i j ) ] , Bo
n = -rij
rij'
f2(x) = __ xv
f l (x) = 2 exp(--x/re) -~ 4x
3xr 2
1 - exp(--x/r¢)
3V '
1 +-- + re
(9)
Further spatial averaging gives
~Ilfv(expl-i~p~E~lldVl =II+lfv(exp(-i~p~E~)-lldVl
~ N. (13)
Introducing the impurity concentration n = N / V and passing to thermodynamic limit N--.0%
185
M.D. Glinchuk et al./Physica B 222 (1996) 182-190
Here we used the fact that X~a = ~ff#~(see Eq. (9)). A~er convertion of hyperbolic functions with imaginary arguments into trigonometric ones with respect to Eq. (15) we have finally from Eq. (16)
V ~ ~, n = const., we obtain (exp I - - i ~ P~E'~I)
/exp I - i =~ Jd=#Pflal - 1 )
=explnfv(eXp(-i~p~E~)-l)dVl.(14)
= cos Q, cos Qr cos Qz - 1 Here we convert the sum over the lattice into an integration. The approximate equality sign in Eq. (13) means that to obtain Eq. (14) from Eq. (12) we neglected higher order (more than pair) correlations of Nb impurities. As it was shown in Refs. [5, 6] this neglect does not play a crucial role for small enough impurity concentrations. Averaging of Eq. (14) over thermal fluctuations of Nb impurities is performed in two steps. First we shall calculate the auxiliary thermal averages (l~) and (l 2) with the one-particle hamiltonian ~ = 1E, I being oriented along the eight directions of [1 1 1] type, E is the argument of the function (12), (l.,r,~)
=
--
( l x , y , z ) = 1, 2
tanh(fE~.r.z); f=(kBT)
(15)
-~
Note that, in spite of the fact that we start from microscopic Nb orientations along the directions of [1 1 1] type, the macroscopic spontaneous polarization orientations in our model are determined by free energy (1) minimization, i.e. the macroscopic behaviour is governed by coefficients B, and C~. We have now from Eq. (14) /exp I - i ~a
~,#P~,I#I) =
cosh (flEx - iQ~) cosh (fE r - iQr ) cosh (fEz - iQ=) cosh (fE~) cosh (tiEr) cosh (fEz)
Oz = Yd~zpx + YCzp~ + YCzzpz.
-
-
(lx) (lz) sin Qx sin Q= cos Qy (ly) (l=) sin Q= sin Qy cos Qx
+ i [ (lx) sin Qx cos Qr cos {2= + (1,.) sin Qycos Q~cos Q~ + (l~) sin Qz cos Qy cos Qx - (lx)(ly)(t~)sinQxsinQysinQ~].
(17) The next step is to obtain of equations for components of average system dipole moment L, = (l,). The latter quantities can be regarded as spontaneous polarization components. Knowledge of them actually permits to determine the equilibrium characteristics of the system like dielectric permittivity, specific heat, etc. (see Refs. [5, 6]). In our approach, however, we shall use the equations for L, to calculate T¢ as the temperature, where L~ # 0 appears. To obtain the equations for L,, we shall use a self-consistent procedure, i.e. express distribution function (11) through itself. In our opinion, the easiest way to do so is to write the equations for L~ in the form L x. r. z =
(lx. r., ) f (E, L) dEx dE r dE=, -
oo
-
ct3
oo
(18)
Q.~ = x ~ p x + YCypy + Ydx=pz; gy = XxrPx + . ~ y P r + ~ff, zPz;
(Ix) (ly) sin Q,, sin Qy cos Qz
-
(16)
where ( l , ) are determined by Eq. (15). f ( E , L ) corresponds to function (11), calculated with respect to Eqs. (14) and (17) but replacing (1,) by L, in Eq. (17). This procedure gives the following final ex-
186
M.D. Glinchuk et al./Physica B 222 H996) 182-190
pression for the distribution function
f(E,t)= 1 fj f°~ f~ exp(iEp)exp[F~(p) oo
--
o~3
--
o0
+ iFz (p)] dpxdprdp~, F1 (p)
n fv [cos Q~ cos Qr cos Qz -
-
Fz(P)
1 fo DpEo(p)exp(Fl(p)) dp, tic = (k,Tc) -1 sinh (np/2fl~)
1= ~
(22)
1
Note that in other words we had shown that the system of Eqs. (18) and (19) has solutions corresponding to tetragonal (Lx = L; L y - - L z = 0, etc), orthorhombic and rhombohedral phases which are determined, however, by a single equation for L [Eq. (19)] and Tc [Eq. (22)]. That is why Eq. (22) with equal probability determines the temperatures of transitions to tetragonal, orthorhombic and rhombohedral phases. It gives the equation for dependence
- L~Ly sin Q~ sin Qycos Qz -
The equation for the phase transition temperature Tc can be obtained from (20) by inserting (21) in the limit L ~ 0. This yields
L~L~ sin Q~sin Qzcos Qr
LyLz sin Qz sin Qr cos Q~] d3r,
n fv [L~ sin Q~cos Qycos Q~ + Lr sin Qy cos Q~ cos Q~
T~ = f (nr3),
+ Lz sin Qz cos Qx cos Qr
(22a)
TcMF -
LxLrLz sin Qx sin Qy sin Q~] d3r, (19)
which determines the temperature and concentrational dependences of the average macroscopic dipole moment of KTN. The Eqs. (18) and (19) can be simplified if we take into account the fact that (Ix) in Eq. (15) depends on Ex only (similarly (ly) and (l~) depend on Er and E,, respectively). So, we can obtain (see Appendix for details) 1 fo ~ sin (F2 (p)) exp (F1 (p)) dp.
L = -fl
(20)
sinh(np/2fl)
The aforementioned fact permits also to simplify the expressions for FI(p) and F2(p). We have
where nr3 is a dimensionless parameter determined by the Nb concentration, TcMF is the phase transition temperature in the mean field approximation. It is determined from the equation (see Refs. [9, 10])
4nnd .2 kBTcMF -- 3e0(T~MF)"
(22b)
As T¢ ~ 0 one can obtain the equation for the Nb critical concentration (nrac)¢rfrom Eq. (22). Let us remind that long-range order occurs in KTN at T = 0 if n > net; in the opposite case, the dipole glass state is realized. The equation for (nrca)~rreads
1 = ~2fo° Eo(p)exp(Fl(p)) dp.
(23)
Fa,2(Px) = FL2(pr) = FL2(p~) = FL2(p), F2(p) = LpEo(p), Fx(p) =
(21)
g o
The numerical solution of this equation with respect to Eq. (9) gives (nr~)~r -~ 0.0184.
(24)
n Jv [cos (Jd~zp) cos (~ffyzP)cos (o~ffzzp)-- 1] d3r,
Eo (p) = ~n f v sin ( ~ 0 ) cos (o~y~p)cos (X~zO)d3r.
The dependence (22a) is shown in the Fig. 2. It will be used in the next section for comparison with experiments on the structural phase transitions in KTN.
M.D. Glinchuk et al./Physica B 222 (1996) 182-190
187
where T is the temperature in Kelvin, G = 6000-8500 cm-2. For our calculations we choose G = 6500 cm- 2. Thus we can rewrite Eq. (22a) in the form
-['c/TcM=
To(x) =
1
10
(nr~)-I
TcMF(X)f(x~(re(~raC))3) ,
(26)
where r~(T~) is determined by Eq. (25). ~ ~- 76.21 is introduced to match the theoretical value (nr3)~r [Eq. (24)] with the experimental one (see Ref. [3]) Xcr ~-- 0.01. The discussion of the reasons why the theoretical value of critical concentration is smaller than the experimental one (this corresponds to = 1 in Eq. (26)) is found in Refs. [5, 6]. The explicit form of equation for T~ME(X)can be obtained from Eq. (22b) by inserting Eqs. (25) and (25a). It reads
Dx
Fig. 2. Dependence of Tc/TcMF on nr 3.
TcMv = 27.1 coth (27.1 / TeMF)
--
12.9'
(27)
where
3. Comparison between theory and experiment; discussion The first step in an adjustment of our theory to the experiment is to obtain the dependence To(x) from Eq. (22a). For the conversion of the parameter nr~ to the atomic concentration x we use the relation
nr3 = x (rc~ 3 ,,a/
(25) '
where a ~- 4 A and r~ are the lattice constants of pure KTaO3 and the correlation radius (see above), respectively; the latter quantity is temperature dependent. We shall calculate the temperature dependence of r~ using the Barrett formula for the dielectric permittivity (or soft mode frequency e)s~) of pure KTaO3 (see Ref. [12]).
t; 0 ~- (D2M '
COs2M--- 26.618 [27.1 coth (~--~) - 12.9] (cm-Z), (25a)
D -~ 5.33 x 108 (Tde)2 ka
(28)
o
Here d (in eA) is the Nb off-center impurity dipole moment; e is the charge of an electron. The constant D can be determined from a well-established experimental point T~(x) at sufficiently large x (x ~_ 0.1), where the mean-field approximation is valid. Such a fit depends on merely one parameter, yd. Since the off-center displacement of Nb is quite well known (see Refs. [13, 14]), this procedure can be considered as that for the experimental determination of, the Lorentz factor 7 at an Nb site. This possibility seems important, because theoretical calculations of 7 (actually 7 = 7(r)) require microscopic models of crystal lattice including consideration of the electronic structure of atoms and ions, interatomic forces, etc (for some details see Refs. [13,14]). It is clear that a model of such kind requires tedious numerical calculations. Obtaining 7 values in such a way depends strongly on the model, which is used for the calculation, so that there is a large difference in their theoretical values. So, in our opinion it is important to start from an independent experimental determination of 7-
M.D.Glinchuketal./PhysicaB222(1996)182-190
188
impurity concentration and other external conditions, which has the smallest free energy. To fulfill this program and to reduce the number of parameters, we shall rewrite the Eqs. (5)-(7) into dimensionless variables. This yields
100.00 @ L.. -,.i-o
U 80.00 L_
.S Z
Q.
E ~)
1
cO •-
x6 '
x 2 = (1 - 0) 1/2 + 1;
60.00
~DR
iiI/~ iiII/~'
40.00
C 0
.~-
20xzR + (2 + r/,)xR4 + 2(1 + r/¢l)x6 ,
2+r/B ( ~1/+ ~c2 - 2(1 + r/c, )
1
40(1 + t/m)'] ; (-2+t/B--~)
L,,.
-@ 20.00 ol 0 _C
o_ 0.00
/ I/
// f
IIl~'h
0.00
iiii1~
zoo
x2 ~ = iitll
~lll
Ilrllllllblllllllll
4oo 6.00 Nb c o n t e n t
Ill~
8.oo
Illll
+ 3(I + r/.)x&
+ 6r/o, +
3(1 + r/.) 3 + 6q¢i + r/¢2
I
o.oo
Fig. 3. Temperatures of transitions to tetragonal (T), orthorhombic (R) and rhombohedral (RE) phases in K T N versus N b content. Solid line: our theory; *: experimental points from Ref. [-3]. Dashed line corresponds to the dependence TCr~v(X).
x ( l + \ / 1 0 ( 3 + 6 r / c~-1--~£-~2l + r / c 2 ) ' ~ - .//, where
Bz r/B =-~1, 0 =
From the experimental value TcMF(X = 0.09) = 95 K [2] we get D = 80 690o K. Hence, dy ~ 0.3 so that o 7 " 3 at d - ~ 0 . 1 e A and 7-~0.6 at d ~ 0.5 cA. The dependence TCMF(X), calculated by solving Eq. (27) with respect to the above obtained D value, is shown in Fig. 3. Now we can calculate Tdx) from Eq. (26). We consider this quantity from the point of view of describing the phase transitions of KTN as temperature of transition from the cubic paraelectric phase, where P = 0, to the first phase, where P ¢ 0, i.e. to tetragonal one. Such supposition is in coincidence with the physical meaning of the equation for T¢ [Eq. (22)]. Let us determine theoretically the temperatures of the structural phase transitions tetragonal-orthorhombic and orthorhombic-rhombohedral. As is usual in Landau phenomenological theory of phase transitions, we shall define these temperatures as those for which free energies of corresponding phases are equal to each other. Note that that phase is stable at a given temperature,
(29)
qgi-
C2,s r/c,.: = C1 '
3~C1
= Oo(x)(T - To(x)),
= 3C1B,
Bal Fi.
Since expression (la) for Z -1 is approximate, dependence Oo(x) cannot be calculated exactly. An approximate form of Oo(x) can be extracted from calculations of Zu "~ OLi/OEj (E is an external electric field), see Ref. [7] for details. Such extraction, made from Z=, calculated in Ref. [7] for a model system with two-orientable dipoles, gives Oo(x) as an expansion into a power series of x. In our case of eight-orientable Nb dipoles it can be shown that Oo(X) will be qualitatively the same. For our purposes it is sufficient to keep terms up to order x 2. The coefficients of this expansion can be extracted for our case from the conditions of the best fit to the experiment with respect to conversion nr~ -+ x (see Eq. (25)). So, we have finally for Oo(X)
Oo(x) ~ 556x 2 - 96.7x + 5.6.
(30a)
This expression together with values r/B ~ 1.65;
qCl ~ 3.5;
r/c2 ~ - 1
(30b)
M.D. Glinchuk et al./Physica B 222 (1996) 182-190
gives a complete determination of the desired free energy. It can be shown that conditions (30a) and (30b) can be satisfied if and only if we put TcR ,~ 0.96T¢;
T¢RE ~ 0.4T¢,
(30c)
where T~R and ToRE are transition temperatures to orthorhombic and rhombohedral phases, respectively, in the case, when the other (i.e. rhombohedral and orthorhombic, respectively) phase would not exist. It means that one has to substitute the values of T~R and TCREto ~0Rand ORE, respectively, instead of T¢ (see Eq. (26)). Introduction of T~R and ToRE have important physical meaning: for TcR < T < T~REwe have long-range tetragonal order with nuclei of orthorhombic phase; these nuclei grow and give rise to the orthorhombic phase at T = TcR; the situation for the rhombohedral phase is similar. It means, that condition (30c), which was obtained from mathematical reasons, coincides with the physical picture of the first-order structural phase transitions in KTN. The calculation of the concentrational dependences of the temperatures of structural phase transitions in K T N can be done as follows. The solution of Eq. (26) with respect to Eq. (27) gives Tc values for each x point. With the help of these values for the same x points we can find the temperatures, where OpT= ~PR and 9R = ~gRE. They are indeed the desired temperatures of structural phase transitions. The theoretical curves Ta(x) are shown in Fig. 3 together with experimental points, taken from Ref. [3]. Good coincidence between theory and experiment is realized. It is also seen that theory differs from experiment for X¢r < X < 0.02. Whereas the experiment shows absence of the structural phase transition sequence in this region (one point instead of three at x > 0.02), theory predicts its presence up to x = x , . The first reason for this discrepancy is the coexistence between the ferroelectric phase and the dipole glass state (see Ref. [15, 16] for details). The second one is the difference in the Nb dipoles reorientation rates (between their eight possible directions) at low and high temperatures (see Ref. [10, 13, 14]). The latter reason results in only one low temperature transition from the paraelectric cubic phase to the rhombohedral ferroelectric one, since the Nb dipoles have to be regarded as non-
189
reorientable, i.e. static. Since we did not account for these reasons, our consideration is valid only at x ~> 0.02, where the structural phase transitions occur.
Note that, for a proper distinction between the dipole glass state and the ferroelectric phase, relaxation measurements are necessary. That is because dielectric relaxation in dipole glasses has long-time character, while in ferroelectrics it is of the conventional Debye type (see, e.g. Refs. [14, 16]). Note also, that the suggested calculation of the KTN free energy in the tetragonal, orthorhombic and rhombohedral phases can be applied for the description of its equilibrium physical properties. It can be used in particular for the description of nonlinear effects in the polarization P(E) and dielectric susceptibility x(E) especially near T --~ T c, where nonlinear effects are essential.
Acknowledgements The research described in this publication was made possible in part by Grants ISSEP SP0042015 and No. U4B200 (M.D.G.) and No. U68000 (V.A.S.) from the International Science Foundation.
Appendix Let us begin with the equation for L [Eq. (18)]. We have for Lx Lx - (27r)3
d3E
tanh (flEx) exp ripE)
x exp [Fl(p) + iF2(p)] d3p.
(A.1)
Integration over d3E in (A.1) can be done as follows
I(p) = ~
tanh(flEx) --or3
--cO
--oC
x exp [i(pxE~ + prEy + pzE~)] dExdEydEz
IA.2)
M.D. Glinchuket al./PhysicaB 222(1996)182-190
190
=(21rt)6(Py)6(Pz)f~o tanh(flEx)sin(pxEx)dEx = For
I(p)
(2re)
~
16(Py)~(pz)"
c a l c u l a t i o n we use the i d e n t i t y exp (ikx) dk = ~(x)
(A.4)
--oO
a n d Ref. [16]. S u b s t i t u t i o n of (A.2) to (A.1) gives Eq. (20) for L - Lx. T h e e q u a t i o n s for Ly,z c a n be d e r i v e d s i m i l a r l y to t h a t for Lx. It c a n be s h o w n t h a t Lx = Ly = Lz.
References [1] D. Sommer, D. Friese, W. Kleemann and D. Rytz, Ferroelectrics 124 (1991) 231. [-2] M.D. Fontana, E. Bouziane and G.E. Kugel, J. Phys.: Condens. Matter 2 (1990) 8681. [3] W. Kleemann, Int. J. Modern Phys. B 7 (1993) 2469. [-4] A.S. Sonin and B.A. Strukov, Vvedenie v segnetoelektrichestvo (Introduction to Ferroelectricity) (Vysshaya Shkola, Moscow, 1970).
[5] V.A. Stephanovich, Free energy functional of disordered dielectrics with dipole impurities, Institute for Theoretical Physics Ukrainian NAS, Kiev, 1994, Preprint ITP-94-62E. [6] V.A. Stephanovich, Ferroelectrics, to be published. [7] B.E. Vugmeister and V.A. Stephanovich, Solid State Commun. 63 (1987) 323. [8] G. Korn and T. Korn, Handbook of Mathematical Functions for Scientists and Engineers (Wiley, New York, 1968). I-9] B.E. Vugmeister and M.D. Glinchuk, Sov. Phys. JETP 52 (1980) 482. [10] B.E. Vugmeister and M.D. Glinchuk, Sov. Phys. Uspekhi 28 (1985) 589. [11] V.A. Stephanovich, Phys. Rev. B 47 (1993) 9084. [12] M.E. Lines and A.M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, Oxford, 1977). [13] B.E. Vugmeister and M.D. Glinchuk, Rev. Mod. Phys. 82 (1990) 993. [14] U.T. H6chli, K. Knorr and A. Loidl, Adv. Phys. 39 (1990) 405. [15] M.D. Glinchuk and V.A. Stephanovich, J. Phys.: Condens. Matter 6 (1994) 6317. [16] M.D. Glinchuk and V.A. Stephanovich, Ferroelectrics 169 (1995) 281. [17] I.S. Gradstein and I.M. Ryzhik, Tables of Integrals, Sums, Series and Multiples (Wiley, New York, 1965).