JOURNAL OF APPLIED PHYSICS
VOLUME 84, NUMBER 1
1 JULY 1998
Kinetics of phase transformations in real finite systems: Application to switching in ferroelectrics Vladimir Shur,a) Evgenii Rumyantsev, and Sergei Makarov Institute of Physics and Applied Mathematics, Ural State University, Ekaterinburg 620083, Russia
~Received 8 December 1997; accepted for publication 23 March 1998! The modification of Kolmogorov–Avrami theory for real objects accounting for the finiteness of the transformed object/media is proposed. It takes into account the changing of the domain growth dimensionality ~‘‘geometrical catastrophe’’! during the switching process. The validity of the proposed approach has been confirmed by model experiments and computer simulation. By this approach we have obtained the essential information about phase kinetics concealed in integrated experimental data. The method has been successfully used for the description of domain kinetics during the fast switching in ferroelectric single crystals and thin films. The scenario of domain evolution and voltage dependence of the main kinetic parameters have been discussed. © 1998 American Institute of Physics. @S0021-8979~98!08712-X#
I. INTRODUCTION
It has been shown that the evolution of ferroelectric domain structure during polarization reversal in electric field ~in bulk single crystals, ceramics, and thin films! is an example of phase transformation.4–6 In this case the local electric field is the driving force of the process. It must be stressed that ferroelectric domain structure is a very appropriate object for the investigation of phase transformation kinetics. First, it is due to the simplicity of managing the domain kinetics by the electric field. Second, the evolution of ferroelectric domains during periodical switching is usually reproducible in details. Third, the possibility of the optical visualization ~with high spatial and time resolution! of instantaneous domain patterns in model single crystals allows to verify the methods of mathematical treatment applied for analyzing the experimental data. The switching data in ferroelectrics were analyzed by Ishibashi and Takagi in the framework of the K–A theory.7 The application of the K–A formula to experimental switching current data in real ferroelectric single crystals,8 ceramics, and thin films9 gives physically unclear values of fitting parameters due to the violations of main postulates of the K–A theory mentioned above. Recently, several attempts to solve the finite-size problem for switching in ferroelectrics were made.10,11 We propose the modification of the K–A theory for the description of phase evolution in real finite systems. The computer simulation and experiments on the model single crystals were used successfully for the verification of the proposed mathematical treatment. The application of the treatment to the fast switching data obtained in thin ferroelectric films allows to distinguish different scenarios of the fast ~and even ‘‘superfast’’! evolution of ‘‘supersmall’’ domains and to determine the voltage dependence of the physical clear kinetic parameters.
The complicated process of first-order phase transitions ~including solid state transformations, crystallization, melting, etc.! occurs by nucleation and growth of the great number of isolated phase volumes. In other words the process of phase transformation is nothing but the evolution of socalled heterophase structure. The understanding of the kinetics of such phase transformations is very important in physics, metallurgy, and materials science. The direct methods which allow to record the instantaneous heterophase patterns during transformation give complete information about time evolution of its geometry and morphology. The application of a commonly used technique such as optical or electron microscopy ~both TEM and SEM! encounters many technical problems, especially in the investigation of the fast processes. Moreover, these direct methods provide the local characteristics of the given region of the sample and need the sophisticated treatment for obtaining the parameters of the phase evolution averaged over the whole volume. On the other hand the universal integrating methods are more simple and popular. The responses obtained in this case are proportional to the instantaneous value of the fraction of daughter/growing phase or its derivative on time. The statistic kinetic parameters are extracted from the integrated experimental data by using the mathematical treatment based on an adequate theoretical description of kinetic process. The statistic theory proposed by Kolmogorov and Avrami ~K–A!1,2 has been widely applied for the mathematical treatment of the data of integrating measurements during phase transformations in various systems, e.g., crystal growth and crystallization of amorphous solids.3 However, the K–A theory being formulated for ideal systems often meets with difficulties while describing the kinetics of transformations in real objects characterized by violations of the main postulates: ~1! finiteness of transformed media and ~2! spatially nonrandom position of nucleation sites.
II. CLASSICAL APPROACH
The K–A statistical theory was developed initially for the crystallization kinetics in metals.1,2 The theory considers
a!
Electronic mail:
[email protected]
0021-8979/98/84(1)/445/7/$15.00
445
© 1998 American Institute of Physics
446
J. Appl. Phys., Vol. 84, No. 1, 1 July 1998
Shur, Rumyantsev, and Makarov
the phase transformation in infinite media under the assumption that the mean size of the individual transformed regions is small as compared to the sample size. It is formulated for the processes with the great number of nuclei randomly distributed over the volume and time. Assuming spatially uniform nucleation probability and the same value of growth velocity for all transforming regions at a given moment the following expression for the time dependence of relative fraction of untransformed/unswitched volume q(t) is obtained:1,2
H
q ~ t ! 5exp 2
Ea z t
0
J
~ ! V @ R ~ z ,t !# d z ,
~1!
where a is the nucleation probability per volume, z is the time arising of the nucleus, V(R)5cR n is the volume occupied by individual transformed region at the moment t, c is the shape constant, R( z ,t) is the radius of the ‘‘individual region,’’ n is the dimensionality of the problem, which can take on the integer values only. Two limiting situations of the nucleation process are usually considered in this model. Following Kolmogorov classification:1 the first, a model is when the nuclei are arising through the whole process with nucleation probability a (t) and, the second, b model is growth of preexisting nuclei, when all nuclei involved in transformation are arising instantaneously at the very beginning of the process with the density b per volume. When the value of the driving force is constant during the whole process the individual region’s growth velocity v and the nucleation probability a are constant. In this case R( z ,t)5 v (t – z ) and the following formulas can be written. For a model: q ~ t ! 5exp@ 2 ~ n11 ! 21 c a v n t n11 # 5exp@ 2 ~ t/t 0 a ! n11 # . ~2! For b model: q ~ t ! 5exp~ 2c b v n t n ! 5exp@ 2 ~ t/t 0 b ! n # .
~3!
The K–A theory was used for the description of the polarization reversal in ferroelectrics in classical works.12,13 The first analysis of the switching current data in ferroelectrics within the K–A theory was presented by Ishibashi and Takagi.7 The switching currents were measured under the action of rectangular electric field pulses with negligible rise time and duration sufficient for complete switching ~Merz technique!.14 The switching current is related to q(t) by the following expression: j ~ t ! 52dq ~ t ! /dt.
~4!
The fitting of the experimental switching current data by using Eqs. ~2!, ~3!, and ~4! with integer n which is the only possibility allowed by the K–A theory (n53 for 3D growth and n52 for 2D growth! is not quite satisfactory. So usually,7–9 the fitting is achieved by varying not only t 0 but n as well. As a result, the noninteger values of the dimensionality n are obtained7–9 @see Fig. 1~b!#. It was proposed7 that the dimensionality of the domain growth can differ from the space dimensionality and depends on the domain shape. n is
FIG. 1. The switching current data measured in PZT/YBCO thin film heterostructure. Experimental points are fitted by theoretical curves: ~a! Kolmogorov–Avrami formula, ~b! Ishibashi–Takagi method, and ~c! proposed finite size modification with geometrical catastrophe.
equal to 3 when the domain boundary moves three dimensionally ~growth in the bulk! @see Fig. 2~c!#. In finite media n52 for the growth of cylinder domains @see Fig. 2~b!#, or n51 for lamella domains @see Fig. 2~a!#. The noninteger values ~obtained during fitting! were interpreted7 as a result of participation of the domain growth processes with different dimensionality. It must be stressed that noninteger value of n is inconsistent with the original approach of the K–A theory. Obtained noninteger values demonstrate that the theoretical postulates are violated in real experimental situations. Moreover, such fitting appears to be ineffective for describing the decaying part of the current pulses @see Fig. 1~b!#.8,9 Similar results can be obtained for any phase transformation in finite media. It is apparent because the K–A theory does not account for the fact that at the instant, when the growing domain touches the boundary of the media, it ceases to grow in one direction. The change of the shape constant occurs. This process significantly changes the time dependence of the growing phase fraction q(t), e.g., switching current shape j(t). It must be stressed that the fitting of the decaying part of switching current is not satisfactory for the processes with the high anisotropy of domain wall motion velocity ~e.g., ferroelectrics–ferroelastics15! or with the ‘‘anisotropic’’ sizes of media ~e.g., thin films!. The influence of the finite size ~so-called finite-grain effect! is very important during switching in polycrystalline films and ceramics.10,11
J. Appl. Phys., Vol. 84, No. 1, 1 July 1998
Shur, Rumyantsev, and Makarov
447
FIG. 3. The scheme demonstrating the geometrical catastrophe 2D–1D for b process in finite media: ~a! anisotropic sizes and ~b! anisotropic growth.
For the simple case of the cube/square shaped media, when V tot5a n : c ~ t ! 5c b 2Dc v t/a, FIG. 2. Scheme demonstrating the variants of domain growth dimensionality: ~a! the growth of the lamella domains in finite media (n51), ~b! the growth of cylinder domains in finite media (n52), ~c! the growth in the bulk (n53).
III. FINITE-SIZE CORRECTIONS
We proposed the modification of the K–A formula for the description of the transformation kinetics in finite media taking into account the above mentioned variation of the shape constant during the process.16–18 For b model and constant value of driving force, q(t) is described by the following expression: q ~ t ! 5exp@ 2 ~ t/t 0 b ! n ~ 12t/t m !# ,
~5!
where the time constant t m accounts for the impingement of growing domains ~the individual transformed regions! on the boundary of the media, t 0 b @see Eq. ~3!#. Let us present the arguments for such modification. At any moment t, it is possible to divide the total volume of the transformed media V tot into two regions: the surface volume V s with the thickness v t ~where v is the growth velocity! and the bulk volume V b 5V tot – V s . All the domains growing from the nuclei situated in the surface volume have touched the boundary to the moment t. The value of their shape constant c s have diminished as regarding its value for the domains growing in the bulk volume c b : c s 5c b 2Dc.
~6!
In this case the following expression can be written for the time dependence of the shape constant averaged over the whole transformed media c ~ t ! 5c b @ 12V s ~ t ! /V tot# 1c s V s ~ t ! /V tot ,
~7!
where V s (t)/V tot is the probability of the nucleus to be found in surface volume at given moment.
~8!
so in this case the time constant t m is t m5
cb a cb t . 5 Dc v Dc cr
~9!
The considered approximation is valid for a/ v @t 0 b , where t 0 b is the time constant of the K–A formula @Eqs. ~3! and ~5!#. We show that the finite-size corrections of the K–A formula for a model can be done in the same manner:18 q ~ t ! 5exp@ 2 ~ t/t 0 a ! n11 ~ 12t/t m !# ,
~10!
where t 0 a @see Eq. ~2#. Equations ~5! and ~10! can only be used for t,t m . Moreover it must be stressed that the proposed consideration is invalid after the moment when all domains have touched the boundary of the media t5t cr ,t m . In isotropic case this formula is always valid during the whole process because the time of complete transformation ~switching time! t s is less than the ‘‘critical time’’ t cr 5a/ v . The anisotropy of growth velocity or sample sizes can modify the considered situation drastically. IV. GEOMETRICAL CATASTROPHES
Let us consider the anisotropic case as defined by two conditions. The first, (t cr ) i 5a i / v i differs sufficiently for various directions of growth i ~strong ‘‘anisotropy of critical times’’! ~see Fig. 3!. The second, the minimal critical time corresponding to the moment when all domains have touched the boundary in one direction is much smaller than the switching time t s : min (t cr ) i !t s . In the anisotropic case the whole process can be divided into stages with different growth dimensionality. At the moment t cat5min(tcr)i ~the time of catastrophe! the dimensionality of the problem jumplike changes ~3D–2D or 2D–1D! ~see Fig. 3!. The final stage of the kinetic process can be described by the same
448
J. Appl. Phys., Vol. 84, No. 1, 1 July 1998
Shur, Rumyantsev, and Makarov
formula ~5! with the reduced integer value of growth dimensionality n→n21. For the b model the experimental data can be fitted by the following expression:
q~ t !5
5
F S D S DG F S D S DG
exp 2
exp 2
n
t
12
t 01
t
n21
t
12
t 02
,
t m1
t
t m2
for 0,t
