Phase-Field Simulation of Ferroelectric Domain Microstructure ...

Materials Transactions, Vol. 50, No. 5 (2009) pp. 970 to 976 Special Issue on Nano-Materials Science for Atomic Scale Modification #2009 The Japan Institute of Metals

Phase-Field Simulation of Ferroelectric Domain Microstructure Changes in BaTiO3 T. Koyama* and H. Onodera Computational Materials Science Center, National Institute for Materials Science, Tsukuba 305-0047, Japan Phase-field method has recently been extended and utilized across many fields of materials science. Since this method can systematically incorporate, the effect of coherency strain induced by lattice mismatch and applied stress as well as external electrical and magnetic fields, it has been applied to many material processes including solidification, solid-state phase transformations, and various types of complex microstructure changes. In this article, we focus on the ferroelectric domain microstructure changes followed by the structural phase transition from cubic to tetragonal phase in BaTiO3 , and its morphological developments are simulated on the basis of the phase-field method. The circuit structure of polarization moments of ferroelectric domains including twin defects is simulated, and the domain morphology is controlled by both the electric dipole-dipole interaction among polarization moments and the elastic interaction among domains with different tetragonal distortion. The ferroelectric domain exchange induced by external electric field is also simulated, then the dielectric property, i.e., the polarization hysteresis curve, is calculated by integrating all the x components of polarization moment vector over the microstructure. Furthermore, the simulation of the reversible ferroelectric domain switching, which is a new phenomenon recently discovered by Ren, is also simulated as an advanced application of the present simulation model. [doi:10.2320/matertrans.MC200806] (Received November 18, 2008; Accepted January 19, 2009; Published April 25, 2009) Keywords: phase-field method, ferroelectric material, polarization domain, polarization hysteresis, time dependent Ginzburg-Landau (TDGL) equation

1.

Introduction

Phase field method has been successfully applied to various materials processes including solidification, solidstate phase transformations and microstructure changes.1–8) Using phase field methodology, one can deal with the evolution of arbitrary morphologies and complex microstructures without explicitly tracking the positions of interfaces. It is rather straightforward to incorporate the effect of coherency and applied stresses, as well as the electrical and magnetic fields. The phase-field simulation method of the structural phase transition in ferroelectric materials and the polarization domain exchanges by external electric field has been proposed by Li et al.,9–11) where the material employed for calculation was the lead zirconate titanate (PZT) thin film. Recently, the temperature-strain phase diagram for BaTiO3 thin films was also calculated based on the phasefield method.12) As the practical application to the ferroelectric random access memory, the ferroelectric domain switching in the memory has been analyzed on the basis of the phase-field model by Dayal and Bhattacharya.13) Therefore, the simulation method has been well established, currently. On the other hand, the reversible ferroelectric domain switching controlled by thermal treatment has been recently discovered by Ren14,15) in BaTiO3 based ferroelectric materials. Since this phenomenon induces the extremely large piezoelectric strain, this phenomenon has been expected as a new type of a piezoelectric actuator. In this paper, firstly, we focus on the ferroelectric domain microstructure changes followed by the structural phase transition from cubic to tetragonal phase in BaTiO3 , and *Corresponding

author, E-mail: KOYAMA.Toshiyuki@nims.go.jp

simulate its morphological developments on the basis of the phase-field method. Then the microstructure development during the ferroelectric domain exchange induced by external electric field and the polarization hysteresis curve with this microstructure changes are simulated. Finally, we modify the conventional simulation method so as to be able to calculate the reversible ferroelectric domain switching newly discovered by Ren,14) and simulate the domain microstructure changes and polarization hysteresis curve during this reversible ferroelectric domain switching. 2.

Simulation Method

In this section, firstly, the phase-field simulation method9–12) of calculating the structural phase transition from cubic to tetragonal phase and the ferroelectric domain microstructure changes in BaTiO3 is explained. Secondary, we account for the mechanism of the reversible ferroelectric domain switching and propose its modeling method based on the phase-field simulation. 2.1

Phase-field simulation of the ferroelectric domain microstructure changes The order parameter describing the ferroelectric domain structure is spontaneous polarization moment Pðr; tÞ ¼ ðP1 ðr; tÞ; P2 ðr; tÞ; P3 ðr; tÞÞ which is the function of spatial position r and time t. This order parameter Pðr; tÞ is regarded as the phase field order parameter in the phase-field method. The temporal evolutions of the polarization Pðr; tÞ, i.e., the structural phase transition and ferroelectric domain microstructure changes, are described by the time dependent Ginzburg-Landau (TDGL) equation: @Pi Gsys ; ¼ LP @t Pi

ð1Þ

Phase-Field Simulation of Ferroelectric Domain Microstructure Changes in BaTiO3

where LP is the relaxation kinetic coefficient and we set LP ¼ 1, which means the time t is normalized to LP and the symbol of normalized time is denoted as t0 in this paper. Gsys is the total free energy of the system, and the quantity Gsys =Pi is the thermodynamic driving force for the spatial and temporal evolution of Pðr; tÞ. The total free energy Gsys is expressed as # Z " 3 1 X 2 Gsys ¼ Gc ðPi ; TÞ þ P jrPi j dr 2 i¼1 ð2Þ r þ Estr þ Edipole þ Eappel ; where Gc ðPi ; TÞ is a bulk free energy density, Estr is an elastic strain energy, Edipole is a dipole-dipole interaction energy among polarization moments in the microstructure, Eappel is an electric potential energy induced from the external electric field, and the second term in the integrant is a ferroelectric domain wall energy which is expressed by the same form as the gradient energy in phase-field method. P is a gradient energy coefficient which is assumed to be constant, P ¼ 8:0 1016 (Jm2 /mol), in this study. We assumed that the strain field and electric field are always at equilibrium for a given polarization field distribution. The bulk free energy density of the BaTiO3 system under a stress-free boundary condition is described by a LandauDevonshire polynomial as Gc ¼ 1 ðP21 þ P22 þ P23 Þ þ 11 ðP41 þ P42 þ P43 Þ þ 12 ðP21 P22 þ P22 P23 þ P23 P21 Þ þ 111 ðP61 þ P62 þ P63 Þ þ 112 ½P21 ðP42 þ P43 Þ þ P22 ðP41 þ P43 Þ þ P23 ðP41 þ P42 Þ þ 123 P21 P22 P23 þ 1111 ðP81 þ P82 þ P83 Þ þ 1112 ½P61 ðP22 þ P23 Þ þ P62 ðP21 þ P23 Þ þ P63 ðP21 þ P22 Þ þ 1122 ðP41 P42 þ P42 P43 þ P43 P41 Þ þ 1123 ðP41 P22 P23 þ P21 P42 P23 þ P21 P22 P43 Þ;

ð3Þ

where the coefficients 1 , ij , ijk and ijkl are adjusted to the BaTiO3 single crystal properties, and the values of which have been determined by Li et al.16) The elastic strain energy is given as Z 1 c 0 Estr ¼ Cijkl f"ijc ðrÞ "ij0 ðrÞgf"kl ðrÞ "kl ðrÞgdr; ð4Þ 2 r where Cijkl is the elastic stiffness tensor, "ijc ðrÞ and "ij0 ðrÞ are the total strain and the stress-free strain, respectively. The elastically homogeneous case and the cubic symmetry are assumed in this study, then the employed values of Cijkl are C11 ¼ 2:48 1011 (Pa), C12 ¼ 1:57 1011 (Pa) and C44 ¼ 0:91 1011 (Pa).17) The stress-free strain "ij0 ðrÞ is expressed by the function of the polarization moment PðrÞ as "011 ðrÞ ¼ Q11 P21 ðrÞ þ Q12 P22 ðrÞ þ Q12 P23 ðrÞ "022 ðrÞ ¼ Q12 P21 ðrÞ þ Q11 P22 ðrÞ þ Q12 P23 ðrÞ "033 ðrÞ ¼ Q12 P21 ðrÞ þ Q12 P22 ðrÞ þ Q11 P23 ðrÞ "012 ðrÞ ¼ ð1=2ÞQ44 P1 ðrÞP2 ðrÞ

ð5Þ

"013 ðrÞ ¼ ð1=2ÞQ44 P1 ðrÞP3 ðrÞ "023 ðrÞ ¼ ð1=2ÞQ44 P2 ðrÞP3 ðrÞ; where coefficients Qij represents the electrostrictive coefficient, and the values of the BaTiO3 single crystal case are

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obtained from Ref. 16). The total strain "ijc ðrÞ is calculated based on the Khachaturyan’s mesoscopic elasticity theory18) as Z 1 "ijc ðrÞ ¼ fpi ðnÞnj nq þ pj ðnÞni nq g k2 ð6Þ dk Cpqlm "0lm ðkÞ expðik rÞ ; ð2Þ3 where k is a reciprocal vector in Fourier space and n k=jkj is unit vector along k direction. ik ðnÞ is the inverse matrix 1 of ik ðnÞ Cijkl nj nl , and "ij0 ðkÞ is the Fourier transform of 0 "ij ðrÞ. The dipole-dipole interaction energy among polarization moments in the microstructure is given as Z Z 3ðri ri0 Þðrj rj0 Þ 1 1 ij Edipole ¼ 2 r r0 4"0 "00 jr r0 j3 jr r0 j5 0 0 Pi ðrÞPj ðr Þdr dr; ð7Þ where ij is Kronecker delta, "0 and "00 are the dielectric constant of vacuum and the relative permittivity, respectively. In this study, we assumed the value of "00 to be "00 ¼ 3000. The electric potential energy induced from the external electric field is calculated by Z Eappel ¼ Ei;appel Pi ðrÞdr ð8Þ r ¼ ðE1;appel P1 þ E2;appel P2 þ E3;appel P3 Þ: Ei;appel is a vector component of external electric field and the Pi is an spatial average of Pi ðrÞ. Since the periodic boundary condition of the microstructure is employed in this study, there is no surface so the depolarization energy is neglected in this study. Numerical simulation of the microstructure changes is performed based on the conventional deference method. 2.2

Phase-field simulation of the reversible ferroelectric domain switching Figure 1 shows the schematic illustration explaining the mechanism of the reversible ferroelectric domain switching, which has been proposed by Ren.14) The light and dark gray regions in Fig. 1(a) represent the ferroelectric domains having tetragonal crystal structure with deferent orientation, respectively. The black arrows in (a) indicate the direction of the c-axis of the tetragonal phase. The small white squares and rectangles in the figure represent the spatial symmetry of the point defects (dopant/impurity ions and Oxygen-ion vacancies) in ferroelectrics, and the square shape means the isotropic spatial distribution of the point defects. On the other hand, the rectangle shape in (b) and (c) indicates that the point defects are preferentially rearranged according to the crystal symmetry of tetragonal phase in which the defects are embedded. Since the defects rearrangement needs short or long range atomic diffusion, this rearrangement is taken place under the thermal treatment such as an aging process. Therefore, Fig. 1(b) shows the situation that the defects arrangement is stabilized by aging. It should be emphasized that the rearranged defects store the orientation of c-axis in its memory through the local atomic bonding or the local strain field. If the external electric field pointing downward direction is imposed on the microstructure of (b), the

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T. Koyama and H. Onodera

(a)

(b)

(c)

Eappel: increase

aging

Eappel : decrease Fig. 1 Schematic illustration explaining the mechanism of the reversible domain switching, which has been proposed by Ren.14) The light and dark gray regions in (a) represent the ferroelectric domains having tetragonal crystal structure with deferent orientation, respectively. The black arrows in (a) indicate the direction of the c-axis of the tetragonal phase. The small white squares and rectangles represent the spatial symmetry of the point defects (dopant/impurity ions and Oxygen-ion vacancies) in ferroelectrics.

ferroelectric domain microstructure will change to (c). When the external electric field is decreased, the ferroelectric domain microstructure will get back to (b), because the previous orientation of c-axis is memorized by the anisotropic point defects arrangement. Further details of this mechanism is explained in Ref. 14). Since this mechanism above mentioned is similar to the mechanism of the rubber-like behavior of reversible martensitic transformation19) as discussed in Ref. 10), we employed the calculation method20,21) proposed for modeling the rubber-like behavior of reversible martensitic transformation to simulate the reversible ferroelectric domain switching in BaTiO3 . In the simulation model of the rubber-like behavior, the new order parameter i ðr; tÞ which describes the slow dynamics of point defects rearrangement is introduced to calculate the reversible domain switching. In addition to eq. (1), the temporal evolution of i ðr; tÞ are also assumed to be described by the time dependent Ginzburg-Landau (TDGL) equation: @i Gsys ; ¼ L @t i

ð9Þ

where L is the relaxation kinetic coefficient of the field i ðr; tÞ. The total free energy of the system Gsys is modified as Z 3 1 X Gsys ¼ jrPi j2 Gc ðPi ; TÞ þ P 2 i¼1 r 3 3 ð10Þ 1 X 1 X 2 2 þ W ðPi i Þ þ ðri Þ dr 2 i¼1 2 i¼1 þ Estr þ Edipole þ Eappel : The third term in the integrant indicates the interaction energy between Pi ðrÞ and i ðrÞ, in which W and are the phenomenological parameters. The fourth term in the integrant is a gradient energy of the field i ðrÞ, and is a corresponding gradient energy coefficient.

3. 3.1

Results and Discussion

Simulation results of the ferroelectric domain microstructure changes Figure 2 shows the two-dimensional simulation of the structural transformation from cubic to tetragonal phase in BaTiO3 at 298 K, where the black part is a cubic phase, and the gray and white parts correspond to the tetragonal phase. The gray level also indicates the orientation of polarization moment vector which is indicated by arrows in Fig. 2(f), the direction of which is parallel to the c-axis of the tetragonal phase. At early stage of structural transformation from cubic to tetragonal phase, the tweed-like structure having the structural modulation along the diagonal direction is developed (see (a)–(c)). Then the microstructure gradually changes to the twin domain morphology during coarsening of domains (see (d)–(f)), where the interface between brighter and darker pert is the twin boundary. This twin microstructure is formed due to relax the elastic constraint induced from the structural phase transition. Furthermore, the continuous circuit of the polarization moments is formed by the dipole-dipole interaction among polarization moments (see the arrows in (f)). Note that the twin domain boundary of the tetragonal phase is coincident with the 90 ferroelectric domain boundary, and the 180 ferroelectric domain boundary is located inside a twin domain. Figure 3 shows the two-dimensional simulation of the polarization domain microstructure change induced by the external electric field applied along horizontal direction (see the upper right arrow). The applied external electric field is set to be constant and the value of which is 4 106 (V/m), and the Fig. 2(f) is employed as the initial microstructure for this simulation, so Fig. 3(a) is the same as Fig. 2(f). The behavior of ferroelectric domain exchanges with the external electric field is simulated from (a) to (f). Initially,


Fig. 2

Two-dimensional simulation of the structural transformation from cubic to tetragonal phase in BaTiO3 at 298 K.

Fig. 3 Two-dimensional simulation of the polarization domain microstructure change induced by the external electric field applied along horizontal direction (see the upper right arrow).

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Fig. 4

Two-dimensional simulation of the polarization domain microstructure changes induced by the external alternating electric field.

the domains having polarization moment along the vertical direction, which is perpendicular to the external electric field, start to disappear (see (b)–(d)), because these domains are energetically unstable not only for the potential energy by the external electric field but also for the elastic strain energy. Then, the domains having polarization moment pointing left hand side shrink, gradually (see (d)–(f)). In this case, since the entire region is a single twin domain, i.e., tetragonal single crystal, the elastic strain energy does not exist. Therefore, the driving force for microstructure changes is only the potential energy induced by the external electric field. Since we confirmed from this result that the ferroelectric domain exchanges is taken place with the external electric field, we further tried to simulate the ferroelectric domain microstructure changes induced by the external alternating electric field. Figure 4 shows the two-dimensional simulation of the polarization domain microstructure changes induced by the external alternating electric field. Since the phase-field method can not calculate nucleation process, the some nucleation site is artificially introduced on the four sides of each figure. The dielectric property, i.e., the polarization hysteresis curve, is calculated by integrating all the x components of polarization moment vector in the microstructure (horizontal direction is x-axis), and the calculated dielectric hysteresis curve corresponding to the microstructure changes in Fig. 4 is represented as Fig. 5, where the abscissa is an external alternating electric field and the vertical axis indicates the average polarization moment along horizontal direction. The solid circles (a) to (h) in Fig. 5 are calculated from the figures (a) to (h) in Fig. 4, respectively. The dielectric hysteresis is reasonably calculated in accordance with the morphological microstructure changes of the ferroelectric domains.

Fig. 5 Calculated dielectric hysteresis curve corresponding to the microstructure changes in Fig. 4.

3.2

Simulation results of the reversible ferroelectric domain switching Figure 6 shows the two-dimensional simulation of the reversible ferroelectric domain switching induced by the external alternating electric field. The initial microstructure morphology of (a) is artificially set up, where the polarization moment Pi ðrÞ takes equilibrium value depending on the morphology of the domain microstructure. The order parameter field i ðrÞ is fixed so as to satisfy the relation i ðrÞ ¼ Pi ðrÞ, which corresponds to the state of the minimum interaction energy (see the third term in the integrant of eq. (10)). Furthermore, since we want to calculate the typical


Fig. 6

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Two-dimensional simulation of the reversible ferroelectric domain switching induced by the external alternating electric field.

case in which the field i ðrÞ strongly constrains the polarization moment field Pi ðrÞ, the parameters W, , and L are assumed to be W ¼ 500 (J/mol), ¼ 1, ¼ 0 and L ¼ 0, respectively. As a result, i ðrÞ does not change from the initial setting. The polarization hysteresis curve is calculated by integrating all the x components of polarization moment vector in the microstructure, and the calculated dielectric hysteresis curve corresponding to the microstructure changes in Fig. 6 is represented as Fig. 7. The circles (a) to (l) in Fig. 7 are calculated from the microstructures (a) to (i) in Fig. 6. The feature of the ferroelectric domain microstructure changes is almost same as that observed in Fig. 3, and the reversible domain switching behavior is calculated, reasonably. 4.

Conclusions

The ferroelectric domain microstructure changes followed by the structural phase transition from cubic to tetragonal phase in BaTiO3 , and the ferroelectric domain microstructure changes are simulated on the basis of the phase-field method. Furthermore, the simulation of the reversible domain switching, which is a new phenomenon recently discovered by Ren, is also simulated as an advanced application of the present simulation model. The results obtained are as follows: (1) The circuit structure of polarization moments of ferroelectric domains including twin defects is simulated, and the domain morphology is controlled by both the electric dipoledipole interaction among polarization moments and the elastic interaction among domains with different tetragonal distortion.

Fig. 7 Calculated dielectric hysteresis curve corresponding to the microstructure changes in Fig. 6.

(2) The ferroelectric domain exchange induced by external electric field is simulated, and the polarization hysteresis curve is calculated, simultaneously, by integrating all the x components of polarization moment vector over the microstructure. (3) The reversible ferroelectric domain switching in BaTiO3 is reasonably modeled by using the calculation method for the rubber-like behavior in reversible martensitic transformation.

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Acknowledgements This study was supported by Grant-in-Aid for Scientific Research on Priority Areas ‘‘Nano Materials Science for Atomic Scale Modification 474’’, and a Next-Generation Supercomputer Project, from Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. The support from a Grantin-Aid Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency (JST) is also acknowledged. REFERENCES 1) T. Koyama: Materia Japan (Bull. the Japan Inst. Metals) 42 (2003) 397. 2) T. Koyama: Ferrum (Bulletin of The Iron and Steel Institute of Japan) 9 (2004) 240. 3) T. Koyama: Ferrum (Bulletin of The Iron and Steel Institute of Japan) 9 (2004) 301. 4) T. Koyama: Ferrum (Bulletin of The Iron and Steel Institute of Japan) 9 (2004) 376. 5) T. Koyama: Ferrum (Bulletin of The Iron and Steel Institute of Japan) 9 (2004) 497.

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