E L S E V I E R Sensors and Actuators B 31 (1996) 115-118 8

e--ff~-m~.

Concentration change influence on the movement of the interphase boundary in Pb(Zr,Ti)O3

A l e x G o r d o n a, S i m o n D o r f m a n b,*, D a v i d F u k s c

aDepartment of Mathematics and Physics, Haifa University at Oranim, 36006 Tivon, Israel bDepartment of Physics, Technion-lsrael Institute of Technology, 32000 Haifa, Israel

CMaterials Engineering Department, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer Sheva, Israel

Abstract

The investigation of the behaviour of the ferroelectric phase transition with concentration change is highly attractive owing to the possibility of preparing alloying samples and to predict theoretically the parameters of the concentration response at relatively small concentrations. These parameters may be extracted from the equation of state of the perovskite under investigation in the assumption of a linear response. The utmost sensitivity of ferroelectric properties to concentration change is the well known from experiments as well as first-principles calculations. One of the most pronounced concentration effects on the ferroelectric properties is the large shift of phase transition temperatures with doping. The study of the movement of the paraelectric-ferroelectric interphase boundary in PB(Zr,Ti)O 3 with concentration change is provided in the framework of the mean-field theory. The analytical solution for the parame- ters of motion of the interphase boundary is applied for the calculations of the width and velocity of the latter at different concentra- tions of Zr. The calculations are based on the experimental data for the Curie-Weiss constant and for the parameters of the Landau- Ginzburg expression for the free energy.

Keywords: Interphase boundary; Perovskite solid solution

Perovskite solid solutions constitute an important group of oxide crystals with broad ranges of technologi- cally important dielectric, piezoelectric, ferroelectric, su- perconducting, and electrooptic properties [1-4]. Some of them have recently attracted attention as very firm ceram- ics, as substances having very large electrooptic coeffi- cients and as materials applied in the development of in- tegrated micromechanical, transistor, memory, and optical devices [1-4]. The static mechanical and electrical prop- erties of the perovskite solid solutions depend strongly on concentration changes of their components. In many of these materials, the ferroelectric phase transitions are first order. As a result of the concentration change of a com- ponent of the solid solution, the first-order phase transi- tion is transformed into second-order at tricritical point [5-9]. At definite concentration range the interphase boundary separating the paraelectric and ferroelectric

* Corresponding author.

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phases exists. Most of the work on ferroelectric perov- skite solid solutions has focused on research of their static properties; here we emphasize the efficiency of investiga- tions of phase transition kinetics due to its high sensitivity to composition changes of the solid solution. We propose a theory of the interphase dynamics at ferroelectric phase transitions in perovskite solid solutions. We calculate the concentration dependences of the width and velocity of the interphase boundary using the experimental data in Pb(Zr,Ti)O3 (PZT) as an example. We show that the ve- locity of the interphase boundary is very sensitive to con- centration changes near the phase transition.

It is well known that concentrated PZT undergoes the same sequence of structural phase transitions as the pure PbTiO3, but at progressively lower phase transition tem- peratures as the concentration of Ti is reduced [5]. At large concentrations of Ti these mixed crystals exhibit the concentration-temperature phase diagram which is similar to the pressure-temperature diagram; the temperature

116 A. Gordon et al. / Sensors and Actuators B 31 (1996) 115-118

ranges of stability of the tetragonal and orthorhombic phases decrease with increasing concentration of Zr and the phase transition temperature T e moves to lower temperatures with increased Zr concentration. There is a concentration at which the character of the transition changes from first-order to second-order. Thus, there is a range of concentrations and temperatures at which the paraelectric-ferroelectric interphase boundary exists. At constant temperature and pressure, the interface exists for concentrations 60 < 6 < 6", where 6" is the concentration corresponding to the stability limit of the ferroelectric phase and 60 is the concentration corresponding to the stability limit of the paraelectric phase; 6* < 6 t, 6 t is the tricritical concentration of Ta. The phase transition con- centration 6 c is the concentration at which the depths of the two minima of the paraelectric and ferroelectric phases are equal. We use here the Landau expansion, which describes well first-order ferroelectric transitions in perovskites:

,7= ![fo +ff~,p2 _1~2P4 +~.~3P6 + ~(Vp)2] df~

(1)

where ~r is the free energy, fo is the free energy density for the paraelectric phase, W is the positive coefficient of the inhomogeneity term. For ~2 and ~3 > 0, Eq. (1) describes a first-order phase transition. For ~2 < 0 a second-order phase transition takes place. If ~2 = 0, a tricritical point is reached ~1 = ~ ( n ) [T-To(n)], where T O is the tempera- ture of the stability limit of the paraelectric phase. It fol- lows from the experimental data [6] that ~ increases linearly with increasing concentration of Zr. The same behaviour is the case for the coefficient ~3 [7],while To and (T c - To) decrease; T e is the phase transition tempera- ture. We can present the coefficient ~2 as follows:

~2 = ~ ( t t - 6 ) (2)

energy density and from the conditions for stability limits of the phases.

By changing the Zr concentration one can induce the motion of the ferroelectric interface. The above dynamics are described in terms of the time evolution of the polari- zation P:

0.__p_P = - F - " (3)

Ot 6P

F is the kinetic coefficient which is assumed to depend non-critically on temperature, pressure and concentration. The functional derivative 6~16P tends to restore the value P to its thermal value. When displaced away from the equilibrium state by changing the concentration of Zr, the system will relax back to it. The kinetics of the relaxation towards equilibrium may be described in terms of the time evolution of the polarization (Eq. (3)). The solution of the kinetic equation (Eq. (3)) for the interface bound- ary conditions has the kink form [ 11 ]

I ¢2.(1+ 1_~i_:~ )

P= 2~3.[ l+exp(_s /A) ] (4)

Here in Eq. (4) we use the substitution s = x - vt, where x is one of the axes. A is the width of the interphase bound- ary given by

1 4" 3llJ'~ 3 A = ~'7" 1 - 2 a + ~

(5)

which moves with the velocity v, given by

V---- F~2 " (8a- 1 - 4 1 - 4 a ) t 2W 3 ~ 3 ( 1 - 2 a + 1-x/~S~

(6)

where ~ does not depend on 6; in the pure PbTiO3, 6 = 0, i.e. ~0 = ~tSt" Eq. (3) is explained as follows. The inter- phase boundary may remain quite sharp unless the con- centration gets sufficiently near the tricritical point, when it becomes dispersed by fluctuations that are large in spa- tial extent. The temperature difference between the phase transition temperature T e and the temperature correspond- ing to the stability limit of the paraelectric phase To de- creases with increasing concentration of Zr and this is a manifestation of the fact that the character of the phase transition changes from first-order to second-order. The dependence D(n) is negligible because W is proportional to the square of the lattice parameter [10] changing neg- ligibly compared to the other composition effects above at the concentration range under consideration [1]. We can obtain 60, 6c and 6* from ao = 0, a c = 3/16, a* = 1/4, where a = ~1~3/~2, i. e. from the minimization of the free

here a = and ~1 ~3/~ , and ~1 is the coefficient renormal- ized by the strain. The kink, described by Eq. (4), is the moving interphase boundary separating the paraelectric (e = 0 at s -+ +oo) and ferroelectric (P = ~/{[~2/2~3][1 + ~/(1 - 4a)]} at s--+--oo) phases. The interphase boundary preserves its shape during the propagation because of the competition of the two terms: the homogeneous part of the free energy density tends to bring the system to a sta- ble state, while the inhomogeneous part of Eq. (3) has the opposite tendency. The interphase boundary moves in the direction of the ferroelectric phase with decreasing Zr concentration in PZT. It moves in the direction of the paraelectric phase with increasing Zr concentration. The interphase boundary velocity v, which is shown in Figs. 1 and 2, as a function of Zr concentration, 6, and overcool- ing temperature, AT, was carried out according to F_x]. (6). The Zr concentration dependence of the velocity, which is

A. Gordon et al. / Sensors and Actuators B 31 (1996) 115-118 117

0.8

0.6

o

0 . 2

0

- 0 . 2 0 0,00.5 0.01 0.015 0.02

concen t ra t i on

Fig. 1. The velocity of the interphase boundary v as a function of Zr concentration ~ in PZT. The plot eorres~=nds to AT = 50°K. The ve- locity is given in units of F~2/3 • (2W/~3) 0"5.

shown in Fig. 1, is calculated for AT= 50°K. The sign of the velocity v defines the direction in which one phase grows at the expense of the other, i.e. the sign depends on the direction in which the interphase boundary propagates thus leading to formation of paraelectric or ferroelectric phases. It is seen from Fig. 1, that a change of 6 up or down to the phase transition point, 6e, induces motion of the interphase boundary. The front is in equilibrium at 6 = 6c, where the free energy densities of the two phases are equal.

In Fig. 2 the phase growth diagram is plotted for the PZT solid solution in coordinates 6 and AT. The behav- iour of the velocity, which is plotted in Fig. 2, results from the competition of the two opposite tendencies of changes of the phase transition temperature. The phase transition temperature decreases when dopant concentra- tion in the sample increases, and it increases with increas- ing overcooling temperature. The intersection of the two surfaces form the curve of phase transitions at which the velocity is equal to zero and the minima of the free en-

O

ergy density have the same depth (see Fig. 2, in which the phase growth diagram is constructed).

A change of 6 above or below t~ C induces the interface motion towards the paraelectric or ferroelectric phase, respectively. This means that the ferroelectric phase grows at the expense of the paraelectric one and vice versa. The curve v(6 - 6e) (6 > 6c) is sharper than v(6 C - 6) (6 < 6e): v o~ (6 - 6c) a, where a = 1.54 and v o~ (6 e -

t~) b, b = 0.92. Exponents a and b are determined by the least square fitting for AT= 50. This asymmetry resem- bles the experimentally observed asymmetry of the tem- perature dependence of the interface velocity on super- cooling and superheating in PbTiO3 [12]. The interface width and velocity are very sensitive to concentration changes. These effects are much stronger than the concen- tration influence on the phase transition temperature. In- deed, the maximal relative change of the phase transition temperature at all the range of concentrations under study is approximately equal to 50%, while at a narrow range of concentrations, e.g. 9-10%, the width increase is ap- proximately equal to 200%. The increase in the concen- tration in 0.2% leads to the 200% growth of the interface velocity at 6 > 6 c. Such a large sensitivity of the velocity to composition changes can provide a new method of studying the solid solutions. By varying the concentration of a component of the perovskite solid solution its growth can be controlled and the kinetics of the first-order phase transition can be governed. Thus, Figs. 1 and 2 can serve as the diagram of growth of the two phases in PZT. Such a diagram can be constructed for each solid solution. The present consideration can be also extended for the study of the kinetic behaviour of other perovskite solid solutions including antiferroelectric ones, for example Ba(Sr,Ti)O 3 [7], K(Ta,Nb)O 3 [6], Co-doped BaTiO 3 [9] because of the analogous concentration dependences of their dielectric properties. It may also be useful for the superconductive perovskite solid solutions undergoing first-order ferroelectric phase transitions. In recent years the approach under study has been investigated experi- mentally and theoretically in the temperature and mag- netic-field induced dynamics of ferroelectric interphase boundaries [12-14]. The measurements on the tempera- ture dynamics are in agreement with the proposed consid- eration [12,13]. In addition, there are measurements on the temperature induced interface dynamics in ferroelec- tric BaTiO3, PbTiO 3, and antiferroelectric NaNbO 3 [15- 20] using the polarization microscope techniques. Such diffusionless interphase boundaries are known in a num- ber of alloys at austenitic-martensitic transformations [21]. The similarity between the two types of phase tran- sitions [22] can be useful in describing the kinetics of martensitic transformations.

Fig. 2. The diagram of the phase growth in the AT- t~ coordinates in PZT. The velocity is given in units of I'~2/3 • (2tla/~3)0"5.

Acknowledgements

The financial support of the Nieders~ichsichen Minis-

118 A. Gordon et al. / Sensors and Actuators B 31 (1996) 115-118

teriums fiJr Wissenschaft und Kultur and the Technion- Haifa University foundation is acknowledged. This re- search was also supported in part by the Israel Ministry of Science and Technology under Grant No. 4868 and in part by the special program of the Israel Ministry of Ab- sorption. Some numerical work was performed at the Florida State University Computer Center.

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