Size dependent domain configuration and electric field driven ...

JOURNAL OF APPLIED PHYSICS 107, 034107 共2010兲

Size dependent domain configuration and electric field driven evolution in ultrathin ferroelectric films: A phase field investigation Yihui Zhang,1 Jiangyu Li,1,2,a兲 and Daining Fang1,3,a兲 1

AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China LTCS, College of Engineering, Peking University, Beijing 100871, China 3 Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195-2600, USA 2

共Received 2 November 2009; accepted 3 January 2010; published online 12 February 2010兲 Size dependent domain configuration and its evolution under an external electric field are investigated for ultrathin ferroelectric films using an unconventional phase field method. The simulation reveals a series of domain configurations at different thicknesses, including zigzag patterns with eight variants or four variants coexisting, a vortex pattern with four variants coexisting, and a stripe pattern with two variants coexisting. When the film thickness falls below a critical value of 2.8 nm, the polarization vanishes, indicating the suppression of ferroelectricity. The evolution of domain configuration under an alternating electric field is also investigated, and the reduction in remnant polarization and coercive field with respect to decreasing thickness is observed. © 2010 American Institute of Physics. 关doi:10.1063/1.3298475兴 I. INTRODUCTION

With ever-increasing demand for device miniaturization and rapid advances in nanoscale fabrication and manipulation of ferroelectrics, low-dimensional ferroelectric materials and devices have received great attentions for their potential applications in microsensors and microactuators,1 nanogenerators,2 phased-array radars,3 and nonvolatile memories.4–6 It is well known that ferroelectricity is a consequence of long-range correlation of electric dipoles, and when the feature size of low-dimensional ferroelectrics approaches the intrinsic correlation length of dipoles,7,8 the collective behavior of dipoles would be substantially different from that in a bulk material, resulting in profoundly different ferroelectric characteristics and properties. As such, the size effect in nanoscale ferroelectrics is of fundamental importance in addition to its technological significance. Furthermore, when the characteristic length of ferroelectrics decreases to nanoscale, the proportion of the surface to volume increases accordingly, leading to great distinctions in atomic configuration of nanoscale ferroelectrics from bulk materials. The surface charges that compensate polarization also become dominant at nanoscale, which could generate a depolarization field large enough to suppress ferroelectricity.9,10 The interplay of the size and surface effects renders lowdimensional ferroelectrics, especially ultrathin ferroelectric films, many new phenomena, and properties, which could be utilized for device applications if they are thoroughly understood. Many experimental and theoretical works have been developed to investigate the size and surface effects in ultrathin ferroelectric films, including their unique polarization distribution, phase transition, electric hysteresis, and the effect of epitaxial strain.9–15 However, it is very challenging to characterize microstructures and macroscopic properties of ultrathin ferroelectric films in situ, making it difficult to dia兲

Authors to whom correspondence should be addressed. Electronic addresses: [email protected] and [email protected].

0021-8979/2010/107共3兲/034107/7/$30.00

rectly correlate the microstructural phenomena in ultrathin ferroelectric films with their macroscopic properties. Piezoresponse force microscopy has recently been developed to characterize domain patterns and local switching characteristics of ferroelectrics simultaneously,16,17 but it only yields a two-dimensional domain structure on the surface, and the local polarization cannot be measured directly. The difficulties in experimental techniques can be overcome if they are complemented by computational investigations, which are able to provide complete microstructural and macroscopic information of ferroelectrics that is difficult to obtain in a single experiment. However, most previous theoretical studies on ultrathin ferroelectric films are based on either univariate Landau–Ginzburg–Devonshire 共LGD兲 theory11,12 that is incapable of analyzing the domain structure in ferroelectrics, or ab initio method9,15 that is not able to deal with large atomic systems. Because of these inherent limitations, neither method can fully reveal the intriguing size and surface effects in ultrathin ferroelectrics. It is well known that the ground state of a ferroelectric material is not uniformly polarized, and ferroelectrics tend to lower their electrostatic energy through the formation of domains. The macroscopic properties of ferroelectrics reflect the collective behavior of domains at micro- and nanoscale, and the configuration and evolution of domain structure have profound influence on the macroscopic behavior of ferroelectrics, especially on their switching characteristics. The interplay of domain structure with the size and surface effects is particularly interesting, resulting in rich microstructural phenomena and intriguing macroscopic properties. Nevertheless, the configuration and evolution of domain structure in ultrathin ferroelectric films remain to be poorly understood. While phase field method based on LGD theory and rigorous variational principle has been widely adopted for domain simulation in ferroelectrics,18–27 its applications in ultrathin

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ferroelectric films have been rarely explored,28,29 and the size and surface effects in low-dimensional ferroelectrics have largely been ignored. In this paper, we develop a computational approach to investigate the configuration and evolution of domain structure in ultrathin ferroelectric films, with particular emphasis on the interplay between the configuration and evolution of domain structure and the size and surface effects. The method extends the recently developed unconventional phase field simulation of ferroelectrics and multiferroics,30–35 which makes it possible to decouple the thickness variation of polarization in a ferroelectric thin film from its in-plane polarization distribution. This is rather difficult to do under a conventional phase field framework, and enables the exploration of size and surface effects on domain configurations in ultrathin ferroelectric thin films, as we show in this paper. II. FORMULATION OF THE PHASE FIELD METHOD

Our key idea to capture the size and surface effects in an ultrathin ferroelectric film is to introduce a new order parameter f共x3兲, which reflects the atomic relaxation along film thickness, so that local spontaneous polarization Pⴱⴱ and transformation strain ␧ⴱⴱ can be expressed in a thicknessdependent form as Pⴱⴱ = f共x3兲Pⴱ and ␧ⴱⴱ = f 2共x3兲␧ⴱ, where Pⴱ and ␧ⴱ are nominal spontaneous polarization and transformation strain of bulk materials. Notice the quadratic variation in transformation strain due to its electrostrictive nature. In conventional phase field theory, Pⴱ is used as order parameters, and the internal energy of the ferroelectric is expanded as a polynomial of Pⴱ.18,19 As a result, the energy well of the ferroelectric is implicit, and the expansion coefficients have to be fine tuned to yield correct material symmetry. To overcome these difficulties, we adopt an unconventional phase field approach proposed by Shu et al.30 instead, and use the characteristic functions of ferroelectric variants as order parameters. For a ferroelectric with N variants, the local spontaneous polarization and transformation strain can be expressed in terms of characteristic function ␥i of variant 共i兲, N

P = 兺 ␥ iP , ⴱ

共i兲

i=1

N

␧ = 兺 ␥ i␧ , ⴱ

共i兲

共1兲

i=1

where P共i兲 and ␧共i兲 are spontaneous polarization and transformation strain of variants 共i兲. Since it is necessary that N ␥i = 1, only N − 1 characteristic functions are indepen兺i=1 dent. To incorporate this constraint, we introduce N − 1 and ␮i, which are independent of each other, and let

␥1 = ␮1 , ␥i = 共1 − ␮1兲 ¯ 共1 − ␮i−1兲␮i,

1 ⬍ i ⬍ N,

␥N = 共1 − ␮1兲 ¯ 共1 − ␮N−2兲共1 − ␮N−1兲,

physical processes, as shown for ferromagnetic shape memory alloys and multiferroic bismuth ferrite.31–33 To capture the domain structure in ultrathin ferroelectric films, we assume that ␮i is independent of x3, implying that the domain walls are perpendicular to film thickness. For an ultrathin film, this assumption is reasonable. The total energy of a ferroelectric film can then be expressed in terms of the order parameters ␮共x1 , x2兲 and f共x3兲 as G共␮, f兲 =

冕

− ␴0 · ␧ − E0 · P兴dx +

␽ 2

冕

R3

兩ⵜ␾兩2dx,

共3兲

where ␴0 and E0 are the applied stress and electric field, and ␽ is the permittivity of free space. In the first integral of the equation, Wg = A1兩ⵜ␮兩2 + A2兩ⵜf兩2 is the gradient energy that penalizes the changes in the order parameters ␮ and f, and thus is interpreted as the energetic cost of forming elastic and N−1 2 ␮i 共1 polar walls separating different variants. Wa = K兺i=1 2 − ␮i兲 is the anisotropy energy of a double-well type, ensuring that the characteristic functions take either 0 or 1. The third term Wdie共f兲 = A3 f 2 represents the internal energy induced by a linear dielectric effect, corresponding to the quadratic term of polarization in the Landau polynomial. The fourth term We共␮兲 = 关␧ − g1共f兲␧ⴱ共␮兲兴 · C · 关␧ − g1共f兲␧ⴱ共␮兲兴 / 2 is the stored elastic energy with C as the elastic modulus and g1共f兲 = f 2共x3兲. The fifth and sixth terms are the potential energies caused by the applied electric field E0 and stress field ␴0. The second integral in Eq. 共3兲 denotes the depolarization energy due to the electric field generated by the polarization distribution in the ferroelectrics. It should be pointed out that the strain field ␧ and the depolarization field Ed = −ⵜ␾ are determined by solving mechanical equilibrium equation and the Maxwell equation subject to proper boundary conditions. The evolutions of the order parameters, ␮共x1 , x2兲 and f共x3兲, are proposed to be governed by the gradient flow of the total energy density G共␮ , f兲 in Eq. 共3兲; i.e.,

⳵␮ = L1共F␮g + F␮a + F␮e + F␮0 + F␮d 兲, ⳵t

共4兲

⳵f 0 e d = L2共Fgf + Fdie f + F f + F f + F f 兲, ⳵t

共5兲

where L1 and L2 are the mobility coefficients, and the driving forces for ␮共x1 , x2兲 are given by F␮g = 2A1ⵜ2␮ , F␮a = −

⳵ W a共 ␮ 兲 , ⳵␮

共2兲

so that the constraint is automatically satisfied. The construction is motivated by the multirank lamination that is proven to be energy minimizing for ferroelectrics,36 and is shown to result in a correct domain structure in bulk ferroelectrics.30 The approach is particularly convenient to couple different

关Wg共ⵜ␮,ⵜf兲 + Wa共␮兲 + Wdie共f兲 + We共␮, f兲

F␮e = g1共f兲

⳵␧ⴱ共␮兲 · C · 关␧ − g1共f兲␧ⴱ共␮兲兴, ⳵␮

F␮0 = fE0 ·

⳵ p ⴱ共 ␮ 兲 , ⳵␮

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F␮d = fEd ·

⳵ p ⴱ共 ␮ 兲 , ⳵␮

共6兲

while those for f共x3兲 are given by Fgf = 2A2 Fdie f = Fef =

F0f =

Fdf =

⳵2 f , ⳵ x23

␧共i兲 = 3␦r共i兲 丢 r共i兲 + 共␣ − ␦兲I,

− 2A3 f ,

1 l20 1 l20 1 l20

冕

⳵ g1共f兲 ⴱ ␧ 共␮兲 · C · 关␧ − g1共f兲␧ⴱ共␮兲兴dx1dx2 , ⳵f x1x2

冕

E0 · pⴱ共␮兲dx1dx2 ,

冕

Ed · pⴱ共␮兲dx1dx2 ,

x1x2

共7兲

x1x2

with l0 being the in-plane dimension of a computational supercell. Notice that the order parameter ␮共x1 , x2兲 satisfies periodic boundary conditions in the x1-x2 plane, while f共x3兲 satisfies the traditional Landau boundary condition;12 i.e., ⳵ f / ⳵x3 兩x3=⫾h/2 = ⫿ f / ␦, with ␦ being the extrapolation length and h being the film thickness, which account for the size and surface effects associated with the ultrathin films. Since the ferroelectric film is widely used as a capacitor operated between two metallic electrodes, the short-circuited electrical boundary condition between the top and bottom surfaces is commonly adopted in both ab initio calculations9,37,38 and phase field simulations.20,28,39,40 In the present simulation, the boundaries are also assumed to be electrically shortcircuited and mechanically clamped in-plane, and thus no out-of-plane depolarization field develops. It should be mentioned that if the open-circuited condition is adopted, the out-of-plane polarization will be smaller than that in the short-circuited case due to the emergence of a depolarization field. The evolution equation of f共x3兲 can be rearranged after proper simplification,

冉

冊

⳵f ⳵2 f = L2 2A2 2 + N1 f + N2 f 3 + F0f + Fdf , ⳵t ⳵ x3 where N1 = − 2A3 + −

2 l20

冕

which has eight ferroelectric variants denoted by r共1兲 = 共111兲, ¯¯1¯1兲, r共3兲 = 共1 ¯ 11兲, r共4兲 = 共11 ¯¯1兲, r共5兲 = 共11 ¯ 1兲, r共6兲 = 共1 ¯ 11 ¯ 兲, r共2兲 = 共1 共7兲 共8兲 ¯ ¯ ¯ r = 共111兲, and r = 共111兲. The transformation strain of the ith variant can be then written as

2 l20

冕

关␧ⴱ共␮兲 · C · ␧兴dx1dx2,

共8兲

N2 =

x1x2

关␧ⴱ共␮兲 · C · ␧ⴱ共␮兲兴dx1dx2 .

共9兲

x1x2

Note that the form of Eq. 共8兲 is similar to that of a traditional univariate LGD equation for ferroelectrics. III. SIMULATION RESULTS AND DISCUSSION A. Rhombohedral ferroelectrics

Now we apply the developed phase field model to study ultrathin ferroelectric films with a rhombohedral structure,

共10兲

where ␣ and ␦ are material constants derived from lattice parameters, r共i兲 is a unit vector along one of the eight 兵111其 crystallographic directions, I is the identity tensor, and the symbol 丢 denotes the tensor product. The material parameters used in the simulations are ␣ = 0 and ␦ = 0.001 31, spontaneous polarization Ps = 0.19 C / m2, and cubic elastic moduli C11 = 194 GPa, C12 = 112 GPa, and C44 = 80 GPa.30 The dielectric effect is taken into account by replacing ␽ in Eq. 共1兲 with ␽␬, and ␬ is the relative dielectric permittivity.21,41 The value of ␬ is taken as 1336 in order to be consistent with the simulation of Shu et al.30 The extrapolation length is taken as ␦ = 20 nm.42 The present formulation involves four additional parameters. Two of them can be grouped into dimensionless forms as D1 = A1 / K / l20 and D2 = A2 / K / l20, and D2 = 100, D1 = 0.1 is taken here. The other two parameters, K and A3, are chosen so that the energy densities Wa, Wdie, and Ws are of the same order. To solve the evolution equations, a semi-implicit finite difference scheme is adopted on a time scale,18–20 and Fourier and finite difference methods are adopted for ␮共x1 , x2兲 and f共x3兲 on a spatial scale, respectively. A three-dimensional 64⫻ 64⫻ 32 supercell is adopted to ensure the accuracy of the calculation, and fast Fourier transformation is adopted to speed up computation in two dimensions 共2D兲. B. Two extreme cases

We first study two extreme cases: one is a very thin rhombohedral film of a few nanometers that has been simulated by molecular dynamics 共MD兲 method,43 and the other is a relatively thick rhombohedral film that approaches behavior of bulk ferroelectrics.30 The in-plane dimension of the supercell is fixed at l0 = 50 nm, while the thickness h varying from 2.4 to 40 nm. Figure 1共a兲 shows the domain configuration obtained from the simulation for film with thickness 2.4 nm⬍ h ⬍ 4.8 nm, where it is observed that the domain exhibits a stripe pattern composed of two variants forming 180° domain walls, which agrees with that calculated from MD simulations,43 as shown in Fig. 1共b兲. Similar stripe domain pattern has also been observed in recent experiments on ultrathin PbTiO3 films.7,44 On the other hand, the domain structure obtained from the simulation of thicker films with h ⬎ 20 nm, when neglecting the out-of-plane depolarization effect, is nearly identical to that obtained by Shu et al.30 for bulk ferroelectrics, as shown in Figs. 1共c兲 and 1共d兲. The zigzag domain pattern appearing in this case has also been observed in PbZr1−xTixO3 共PZT兲 in the vicinity of a morphotropic phase boundary,45 and in a rhombohedral Pb共Mg1/3Nb2/3兲O3 – PbTiO3 crystal46 where the etched sample exhibits periodic prominence and depression corresponding to the upward and downward polar directions. These two sets of simulations in the extreme cases thus validate the phase field method we developed in this paper. In

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FIG. 3. 共Color online兲 The size dependent out-of-plane polarization and domain pattern. The domain exhibits a zigzag pattern with eight variants coexisting when h ⬎ 7.6 nm, a zigzag pattern with four variants coexisting when 6.8 nm⬍ h ⬍ 8.0 nm, a vortex pattern with four variants coexisting when 4.4 nm⬍ h ⬍ 7.2 nm, and a stripe pattern with two variants coexisting when 2.4 nm⬍ h ⬍ 4.8 nm; when the thickness is smaller than 2.8 nm, the domain disappears, indicating that the ferroelectricity is suppressed. FIG. 1. 共Color online兲 Two extreme cases simulated by the present phase field method: 共a兲 the domain pattern of an ultrathin film calculated by the present phase field method with 2.4 nm⬍ h ⬍ 4.8 nm, 共b兲 the domain pattern of the ultrathin film calculated by MD with h = 4.4 nm 共Ref. 39兲, 共c兲 the domain pattern of a thin film calculated by the present phase field method with h ⬎ 20 nm without considering the out-of-plane depolarization and 共d兲 the domain pattern of the thin film calculated by the 2D phase field simulation without concerning the size effect 共Ref. 30兲. The symbol 䉺 共 丢 兲 denotes the direction of polarization flowing out of 共into兲 the x1x2 surface.

the following investigation, this phase field method is employed to analyze the size effect in ultrathin ferroelectric films. C. Size dependent spontaneous domain configurations

We first consider the variation of average out-of-plane polarization along the thickness direction for films with different thicknesses, as shown in Fig. 2. The distributions are

FIG. 2. 共Color online兲 The size dependent distribution of the out-of-plane polarization at different thicknesses.

almost similar for all films, with a larger polarization inside and a smaller one outside, which is consistent with the prediction by traditional univariate LGD theory.11,12 Since the average value of the polarization is relatively small for a 2.8 nm film, the slope is also very small as can be observed from the boundary condition. It can be also found that the thinner film has a smaller average polarization and a more uniform polarization distribution. While the traditional LGD theory11,12 is only capable of investigating the size dependence of out-of-plane polarization, our method is also effective in analyzing the size dependence of domain configuration. Figure 3 shows the variation of the domain pattern and the average out-of-plane polarization with respect to the film thickness. With decreasing film thickness, the domain configuration undergoes three transitions: from a zigzag pattern with eight variants’ coexistence to a zigzag pattern with four variants’ coexistence at h = 7.6 nm, and then to a vortex pattern with four variants’ coexistence at h = 6.8 nm, and finally to a stripe pattern with two variants’ coexistence at h = 4.4 nm. When the film thickness falls further below 2.8 nm, the domain pattern vanishes and the polarization becomes zero, indicating the suppression of ferroelectricity for films thinner than 2.8 nm. This critical thickness of 2.8 nm is similar to that predicted by ab initio calculations9 for BaTiO3 films between two metallic SrRuO3 electrodes. It can also be found from Fig. 3 that the number of coexisting variants decreases with the thickness. This decrease in coexisting variants is due to the competition between the gradient energy Wg, which favors a single-domain state, and the elastic energy We and the depolarization energy Wd, which favor domain formation. Both elastic and depolarization energies decrease monotonically with the magnitude of polarization while the gradient energy is independent of it. Therefore, with decreasing film thickness, the gradient energy becomes increasingly more important, and the variant number decreases as a result to reduce the gradient energy.

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FIG. 4. 共Color online兲 共a兲 The zigzag domain pattern with eight variants coexisting when h ⬎ 7.6 nm. 共b兲 The corresponding in-plane polarization distribution. 共c兲 The initiation of a vortex near the domain wall.

For a thicker film, the zigzag domain pattern with eight variants’ coexistence is shown in greater detail in Fig. 4共a兲, which is similar to a domain pattern in bulk ferroelectric 关Fig. 1共c兲兴, though domain walls become smoother. Three types of domain walls are observed, including 180°, 109°, and 71°, as shown in Figs. 4共a兲 and 4共b兲. A magnification of the in-plane polarization distribution demonstrates that a vortex polarization structure appears near the 180° domain wall, as shown Fig. 4共c兲. For modest film thickness with the zigzag domain pattern and four variants’ coexistence, the inplane polarization distribution is similar to that displayed in Fig. 4共b兲, with the exception that only the 180° domain wall and the 109° domain wall exist in this case. For the vortex domain pattern in a thinner film, it is discovered that two lines of vortex polarization structures along ⫾45 directions exist, with clockwise and anticlockwise vortices arranged alternatively, as shown in Fig. 5. Similar vortex polarization structure has been observed in ab initio simulations of a PZT nanowire with diameter d = 2.8 nm.47 Only the 109° and 71° domain walls exist in this polarization structure. For the thinnest film with a stripe domain pattern, only two variants and a 180° domain wall exist.

FIG. 5. 共Color online兲 The vortex structure of polarization in ultrathin films predicted by our phase field simulations.

FIG. 6. 共Color online兲 The size dependent electric hysteresis loop of rhombohedral ferroelectric films.

D. Size dependent hysteresis loops and domain evolutions

One of the defining characteristics of ferroelectrics is their hysteresis loop, which resulted from the evolution of domain configuration. Due to the size dependent domain pattern and spontaneous polarization, the hysteresis loop of the ultrathin film is expected to exhibit different characteristics from the bulk ferroelectrics. To examine this effect, we use the equilibrium domain structure obtained in the previous simulation as the initial state, and an alternating cyclic electric field along x3 is applied to pole and depole the film periodically. As shown in Fig. 6, the hysteresis loop becomes smaller and narrower with decreasing thickness, indicating that both the remnant polarization and the coercive field decrease with the film thickness. This is consistent with recent experiments on ultrathin ferroelectric PbTiO3 films48,49 as well as MD simulations on ultrathin ferroelectric BaTiO3 films.43 A further investigation reveals that the remnant polarization satisfies a near-linear relation with the reciprocal of film thickness 共1 / h兲, as displayed in Fig. 7. We then take the ultrathin film with thickness h = 20 nm as an example to illustrate the domain evolution during the poling and depoling processes. With the increase

FIG. 7. The size dependent remnant polarization of rhombohedral ferroelectric films as a function of the reciprocal of the film thickness.

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ACKNOWLEDGMENTS

The authors are grateful for the support from the Natural Science Foundation of China under Grant Nos. 10572069, 10820101048, and 10732100. O. Auciello, J. F. Scott, and R. Ramesh, Phys. Today 51共7兲, 22 共1998兲. Z. L. Wang and J. H. Song, Science 312, 242 共2006兲. 3 F. W. Van Keuls, N. C. Varaljay, C. H. Mueller, S. A. Alterovitz, F. A. Miranda, and R. R. Romanofsky, Integr. Ferroelectr. 77, 51 共2005兲. 4 J. F. Scott, Ferroelectric Memories 共Springer, Berlin, 2000兲. 5 J. F. Scott, Science 315, 954 共2007兲. 6 L. Liao, H. J. Fan, B. Yan, Z. Zhang, L. L. Chen, B. S. Li, G. Z. Xing, Z. X. Shen, T. Wu, X. W. Sun, J. Wang, and T. Yu, ACS Nano 3, 700 共2009兲. 7 D. D. Fong, G. B. Stephenson, S. K. Streiffer, J. A. Eastman, O. Auciello, P. H. Fuoss, and C. Thompson, Science 304, 1650 共2004兲. 8 I. I. Naumov and H. Fu, Phys. Rev. Lett. 95, 247602 共2005兲. 9 J. Junquera and P. Ghosez, Nature 共London兲 422, 506 共2003兲. 10 D. J. Kim, J. Y. Jo, Y. S. Kim, Y. J. Chang, J. S. Lee, J. G. Yoon, T. K. Song, and T. W. Noh, Phys. Rev. Lett. 95, 237602 共2005兲. 11 J. F. Scott, H. M. Duiker, P. D. Beale, B. Pouligny, K. Dimmler, M. Parris, D. Butler, and S. Eaton, Physica B 150, 160 共1988兲. 12 W. L. Zhong, Y. G. Wang, P. L. Zhang, and B. D. Qu, Phys. Rev. B 50, 698 共1994兲. 13 K. J. Choi, M. Biegalski, Y. L. Li, A. Sharan, J. Schubert, R. Uecker, P. Reiche, Y. B. Chen, X. Q. Pan, V. Gopalan, L. Q. Chen, D. G. Schlom, and C. B. Eom, Science 306, 1005 共2004兲. 14 C. H. Ahn, K. M. Rabe, and J. M. Triscone, Science 303, 488 共2004兲. 15 M. Stengel, D. Vanderbilt, and N. A. Spaldin, Nature Mater. 8, 392 共2009兲. 16 S. V. Kalinin and D. A. Bonnell, Phys. Rev. B 65, 125408 共2002兲. 17 M. J. Brukman and D. A. Bonnell, Phys. Today 61共6兲, 36 共2008兲. 18 L. Q. Chen, Annu. Rev. Mater. Res. 32, 113 共2002兲. 19 Y. L. Li, S. Y. Hu, Z. K. Liu, and L. Q. Chen, Appl. Phys. Lett. 78, 3878 共2001兲. 20 Y. L. Li, S. Y. Hu, Z. K. Liu, and L. Q. Chen, Appl. Phys. Lett. 81, 427 共2002兲. 21 S. Choudhury, Y. L. Li, C. E. Krill III, and L. Q. Chen, Acta Mater. 53, 5313 共2005兲. 22 J. Wang, S. Q. Shi, L. Q. Chen, Y. L. Li, and T. Y. Zhang, Acta Mater. 52, 749 共2004兲. 23 J. Wang, Y. L. Li, L. Q. Chen, and T. Y. Zhang, Acta Mater. 53, 2495 共2005兲. 24 K. Dayal and K. Bhattacharya, Acta Mater. 55, 1907 共2007兲. 25 Y. Su and G. J. Weng, J. Mech. Phys. Solids 53, 2071 共2005兲. 26 Y. C. Song, A. K. Soh, and L. Lu, J. Phys. D: Appl. Phys. 41, 082002 共2008兲. 27 L. Hong, A. K. Soh, S. Y. Liu, and L. Lu, J. Appl. Phys. 106, 024111 共2009兲. 28 J. Wang and T. Y. Zhang, Phys. Rev. B 73, 144107 共2006兲. 29 L. Hong, A. K. Soh, Y. C. Song, and L. C. Lim, Acta Mater. 56, 2966 共2008兲. 30 Y. C. Shu, J. H. Yen, H. Z. Chen, J. Y. Li, and L. J. Li, Appl. Phys. Lett. 92, 052909 共2008兲. 31 L. J. Li, J. Y. Li, Y. C. Shu, and J. H. Yen, Appl. Phys. Lett. 93, 192506 共2008兲. 32 Y. F. Ma and J. Y. Li, Acta Mater. 55, 3261 共2007兲. 33 Y. F. Ma and J. Y. Li, Appl. Phys. Lett. 90, 172504 共2007兲. 34 Y. C. Shu and J. H. Yen, Appl. Phys. Lett. 91, 021908 共2007兲. 35 Y. C. Shu and J. H. Yen, Acta Mater. 56, 3969 共2008兲. 36 J. Y. Li and D. Liu, J. Mech. Phys. Solids 52, 1719 共2004兲. 37 M. Nuñez and M. B. Nardelli, Phys. Rev. Lett. 101, 107603 共2008兲. 38 T. Nishimatsu, U. V. Waghmare, Y. Kawazoe, and D. Vanderbilt, Phys. Rev. B 78, 104104 共2008兲. 39 Y. L. Li, S. Y. Hu, and L. Q. Chen, J. Appl. Phys. 97, 034112 共2005兲. 40 S. Choudhury, Y. L. Li, L. Q. Chen, and Q. X. Jia, Appl. Phys. Lett. 92, 142907 共2008兲. 41 J. Wang and T. Y. Zhang, Appl. Phys. Lett. 88, 182904 共2006兲. 42 J. W. Hong and D. N. Fang, J. Appl. Phys. 104, 064118 共2008兲. 43 Y. L. Sang, B. Liu, and D. N. Fang, Comput. Mater. Sci. 44, 404 共2008兲. 44 C. Thompson, D. D. Fong, R. V. Wang, F. Jiang, S. K. Streiffer, K. Latifi, J. A. Eastman, P. H. Fuoss, and G. B. Stephenson, Appl. Phys. Lett. 93, 182901 共2008兲. 45 T. Asada and Y. Koyama, Phys. Rev. B 75, 214111 共2007兲. 1 2

FIG. 8. 共Color online兲 The domain evolution under an alternating cyclic electric field, and 共a兲–共h兲 correspond to the eight points a–h in Fig. 6: 共a兲 E3 = 0 ⫻ 106 V / m, 共b兲 E3 = 0.5⫻ 106 V / m, 共c兲 E3 = 0.6⫻ 106 V / m, 共d兲 E3 = 0.7⫻ 106 V / m, 共e兲 E3 = −3.2⫻ 106 V / m, 共f兲 E3 = −3.3⫻ 106 V / m, 共g兲 E3 = −3.4⫻ 106 V / m, and 共h兲 E3 = −3.5⫻ 106 V / m.

in the electric field from E3 = 0 to E3 = 0.6⫻ 106 V / m 关from Figs. 8共a兲–8共c兲兴, four 71° domain walls all start moving to increase the positive out-of-plane polarization. As the electric field increases further to E3 = 0.7⫻ 106 V / m 关from Figs. 8共c兲 and 8共d兲兴, the domain structure experiences a drastic change from a zigzag pattern into a stripe pattern with only ¯¯11兲, suggesting that two variants polarized along 共111兲 and 共1 the ferroelectric film is fully saturated. After the reversal of the electric field, the nucleation of new domain does not occur until the field decreases to about E3 = −3.2 ⫻ 106 V / m, as shown in Fig. 8共e兲. The new domain polar¯¯1¯1兲 direction nucleates from the ized along the 共1 ¯ ¯ 共111兲-polarized domain and grows larger with a further decrease in the field to E3 = −3.4⫻ 106 V / m, as shown from Figs. 8共e兲–8共g兲. When the electric field decreases below the coercive field, the original domains polarized along the 共111兲 ¯¯11兲 directions both switch to be along the −x direcand 共1 3 tion, leading to the formation of the stripe pattern shown in Fig. 8共h兲. For thinner films with other type of domain structure shown in Fig. 3, the saturated domain configurations also exhibit the stripe pattern with two variants coexisting, and thus the domain evolutions during the polarization reversal process are similar to that of a 20 nm thick film as shown from Figs. 8共d兲–8共h兲, which will not be repeated here. The smaller coercive field for a thinner film is mainly due to the reduced remnant polarization as shown in Fig. 7.

IV. CONCLUSIONS

In conclusion, an unconventional phase field method has been developed to investigate the size dependence of domain configuration and evolution in ultrathin ferroelectric films. The simulations reveal that four types of compatible domain structures exist at different thicknesses, with the number of coexisting variants decreasing with the film thickness. It is also observed that both remnant polarization and coercive field decrease as film thickness is reduced, resulting in a smaller hysteresis loop at smaller thickness, and eventually suppression of ferroelectricity below a critical thickness.

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Zhang, Li, and Fang

M. C. Shin, S. J. Chung, S. G. Lee, and R. S. Feigelson, J. Cryst. Growth 263, 412 共2004兲. 47 I. I. Naumov, L. Bellaiche, and H. X. Fu, Nature 共London兲 432, 737 共2004兲. 48 Y. S. Kim, D. H. Kim, J. D. Kim, Y. J. Chang, T. W. Noh, J. H. Kong, K.

J. Appl. Phys. 107, 034107 共2010兲 Char, Y. D. Park, S. D. Bu, and J. G. Yoon, Appl. Phys. Lett. 86, 1029 共2005兲. 49 V. Nagarajan, J. Junquera, J. Q. He, C. L. Jia, R. Waser, K. Lee, Y. K. Kim, S. Baik, T. Zhao, R. Ramesh, Ph. Ghosez, and K. M. Rabe, J. Appl. Phys. 100, 051609 共2006兲.