APPLIED PHYSICS LETTERS 93, 262904 共2008兲

Domain control in ferroelectric nanodots through surface charges Jie Wanga兲 and Marc Kamlah Forschungszentrum Karlsruhe, Institute for Materials Research II, Postfach 3640, 76021 Karlsruhe, Germany

共Received 17 October 2008; accepted 7 December 2008; published online 29 December 2008兲 Stable polarization distributions of freestanding ferroelectric nanodots with different surface charges are investigated numerically using a phase field model. The out-of-plane components of polarizations are found to be proportional to the density of surface charge. When the density of surface charge exceeds a critical value, the in-plane components of polarizations disappear. It makes ferroelectric nanodots experience an unusual transition from a vortex state to a single-domain state. Simulation results also show that regular multidomain structures can be obtained by means of specified surface charges, which suggests a way to tailor the physical properties for specific applications. © 2008 American Institute of Physics. 关DOI: 10.1063/1.3058821兴 The physical properties of ferroelectric materials, such as the piezoelectric, dielectric, and photonic properties, are highly dependent on the polarization distribution or domain structures in the materials.1 The ability to control the domain orientation in ferroelectric nanostructures holds promise as a method to tailor physical properties for specific applications.2 Stable polarization distributions in ferroelectric nanostructures are mainly determined by the competition between different long-range interactions under specified boundary conditions.3,4 Under an ideal short-circuit boundary condition, single-domain structures are stable in ferroelectric nanodots4 and nanotubes3 since the polarizationinduced surface charges are completely compensated by external charges. Under an open-circuit boundary condition without any charge compensation, vortex structures of polarization are energetically favorable in ferroelectric nanostructures.5 In typical ferroelectric devices, the surface charges are compensated by two metallic electrodes sandwiching the ferroelectric materials. Recent experimental and theoretical investigations show that the screening of surface charge by atomic and molecular adsorbates is more effective than metallic and oxide electrodes in stabilizing the polarization in ferroelectric nanostructures.6 This compensation mechanism allows for a further miniaturization of ferroelectric devices. How the surface charge influence the polarization distribution of ferroelectric nanostructures is still not well understood. On the other hand, surface charges can be used to control the domain patterns in ferroelectric nanostructures.7,8 Therefore, the prediction of equilibrium domain structures in ferroelectric nanostructures with different surface charges is not only important in physics but also technically useful. In this letter, we take freestanding PbTiO3 nanodots as an example to investigate how the surface charge changes the equilibrium polarization distribution in ferroelectric nanostructures using a phase field model. The phase field model is the same as the one in our previous work.3 Similar phase field models were extensively employed to predict the equilibrium polarization distribution in ferroelectric thin films and bulk materials.9–14 In the present phase field model, a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]. Tel.: ⫹49 7247 82 5857. FAX: ⫹49 7247 82 2347.

0003-6951/2008/93共26兲/262904/3/$23.00

the total free energy of the ferroelectric system is obtained by integrating an electrical enthalpy over the whole volume. The electrical enthalpy is a function of polarization Pi, strain ␧ij, polarization gradient Pi,j 共=⳵ Pi / ⳵x j兲, and electric field Ei, which can be expressed as15–17 h共Pi,␧ij, Pi,j,Ei兲 = ␣i P2i + ␣ij P2i P2j + ␣ijk P2i P2j P2k + 21 cijkl␧ij␧kl − qijkl␧ij Pk Pl + 21 gijkl共⳵ Pi/⳵ x j兲共Pk/⳵ xl兲 − 21 ␧0EiEi 共1兲

− Ei Pi ,

in which the repeated indices imply summing over 1, 2, and 3. The first three terms in Eq. 共1兲 represent the Landau– Devonshire free energy, where ␣i = 共T − T0兲 / 2␧0C0 is the dielectric stiffness, ␣ij and ␣ijk are higher order dielectric stiffnesses, T and T0 denote the temperature and the Curie-Weiss temperature, respectively, C0 is the Curie constant, and ␧0 is the dielectric constant of vacuum. The fourth term denotes the elastic energy of the system, in which cijkl is the elastic constant. The fifth term is the coupling energy between polarizations and strains, where qijkl is the electrostrictive coefficient. The term of 21 gijkl共⳵ Pi / ⳵x j兲共Pk / ⳵xl兲 is the gradient energy, in which gijkl is the gradient coefficient. The last two terms are the electrical energy. With the electrical enthalpy at hand, the stresses and electric displacements can be derived as ␴ij = ⳵h / ⳵␧ij and Di = −⳵h / ⳵Ei. The temporal evolution of the polarization formation in ferroelectrics can be obtained from the following timedependent Ginzburg–Landau equation

⳵ Pi共x,t兲 ␦F =−L ␦ Pi共x,t兲 ⳵t

共i = 1,2,3兲,

共2兲

where L is the kinetic coefficient, F = 兰Vhdv is the total free energy of the simulated system, ␦F / ␦ Pi共x , t兲 represents the thermodynamic driving force for the spatial and temporal evolution of the simulated system, x denotes the spatial vector x = 共x1 , x2 , x3兲, and t is time. In addition to Eq. 共2兲, the mechanical and electrical equilibrium equations of ␴ij,j = 0 and Di,i = 0 must be simultaneously satisfied for charge-free and body-force-free ferroelectric materials under the boundary conditions of ␴ijn j = 0 and Dini = ␻, respectively, in which the comma represents the partial derivation with respect to

93, 262904-1

© 2008 American Institute of Physics

Downloaded 06 Jan 2009 to 117.32.153.167. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

262904-2

J. Wang and M. Kamlah

FIG. 1. 共Color online兲 Polarization distributions in the ferroelectric nanodots without surface charge 共a兲–共c兲 and in the ferroelectric nanodots with surface charge 共d兲. 共a兲 Three-dimensional spatial distribution of polarization magnitudes in the ferroelectric nanodot. 共b兲 Two-dimensional projection of the three-dimensional polarization vectors in the x1-x2 plane at x3 = 2 nm. 共c兲 Two-dimensional projection of the three-dimensional polarization vectors in the x1-x3 plane at x2 = 8 nm. 共d兲 The same as 共c兲 but with uniformly distributed positive charges on the upper surface and the equivalent negative charges on the lower surface.

the coordinate, n j denotes the component of unit vector normal to the surface, and ␻ is the charge density on the surface. A nonlinear multifield coupling finite element formulation is employed to discretize the governing Eq. 共2兲, ␴ij,j = 0 and Di,i = 0 at each time step. The materials parameters used in the simulation are listed in Ref. 18, which are converted from constant-stress to constant-strain coefficients.11,19,20 The spontaneous polarization is 0.76 C / m2 based on the material parameters of Ref. 18, and the corresponding spontaneous strains are ␧33 = 5.1% and ␧11 = ␧22 = −1.5% if the spontaneous polarization is along the x3 direction. In the simulation, we employ 4000 discrete brick elements to model ferroelectric nanodots with 20 elements in the x1 and x2 directions, ten elements in the x3 direction. The dimensions of the simulated ferroelectric nanodots are 8, 8, and 4 nm in the x1, x2, and x3 directions, respectively. The free boundary condition of ⳵ P / ⳵n = 0 is used for the polarizations on the surfaces.21 The backward Euler scheme is adopted for the time integration in Eq. 共2兲. At each time step, the nonlinear equations are solved by Newton iteration. At the beginning of the evolution, a random fluctuation of polarization field is introduced to initiate the polarization evolution process.10,11 In the present letter, only results at steady state and at room temperature are presented. In the absence of surface charges, the simulated ferroelectric nanodot forms a pure vortex structure, in which the vortex axis coincides with the x3 axis. Figure 1共a兲 shows the three-dimensional spatial distribution of polarization magnitude of the ferroelectric nanodot. The contour legend shows different magnitudes of polarizations in different colors. Figure 1共b兲 gives the two-dimensional projection of the threedimensional polarization vectors in the middle plane of the nanodot at x3 = 2 nm. The origin point of the x3 axis is located at the lower surface of the dot in Fig. 1共a兲. It is found that the polarizations form into a closed vortex pattern in the x1-x2 plane and have a head-to-tail arrangement, which

Appl. Phys. Lett. 93, 262904 共2008兲

FIG. 2. 共Color online兲 Polarizations vs the surface charge density. The inplane components 具兩P1兩典 and 具兩P2兩典 become zero when the normalized charge density exceeds a critical value of 1.0. The relationship between the out-ofplane component and the charge density is found to be 具兩P3兩典 = ␻ through linear fitting.

greatly reduces the depolarization energy. Nevertheless, there is still an internal depolarization field induced by inhomogeneous polarizations inside the dot according to the Gauss’ law. The polarization components in the x3 direction are zero in the absence of surface charges. This can be confirmed by Fig. 1共c兲, which shows the two-dimensional projection of the three-dimensional polarization vectors in the x1-x3 plane at x2 = 8 nm. However, the polarization components in the x3 direction are nonzero when there are external fixed charges on the surfaces as shown in Fig. 1共d兲, in which the normalized charge density is 0.5. Figure 1共d兲 gives the twodimensional polarization vectors on the surface of x2 = 8 nm when the dot has uniformly distributed positive charges on the upper surface and the equivalent negative charges on the lower surface. Figure 2 shows the change in polarization components when the charge density increases. The quantities in Fig. 2 are the averages of the absolute value of the component at all nodes. The polarization components in the x1-x2 plane form a pure vortex structure, which makes the averages of the inplane components, i.e., 具P1典 and 具P2典, to be zero. To represent the in-plane components existing inside the dots, we first take the absolute values of the components and then take an average over them. It is found that the out-of-plane component, i.e., 具兩P3兩典, increases linearly with the charge density regardless of the value as shown by the line with solid circle in Fig. 2. The relationship between the out-of-plane component and the charge density is found to be 具兩P3兩典 = ␻ through linear fitting. The in-plane components 具兩P1兩典 and 具兩P2兩典 are the same for all charge densities as shown by the two curves with solid square and star in Fig. 2. When the surface charge density is small, 具兩P1兩典 and 具兩P2兩典 have almost the same value as for the case without surface charge. When the normalized surface charge density increases to 0.875, the in-plane polarizations become smaller. The polarizations near the four corners and in the center first become zero while the nonzero polarizations still form a vortex structure similar to Fig. 1共b兲. When the normalized surface charge density increases to 1.0, all the in-plane polarizations become zero and the dots change from the vortex state to the homogeneous state as shown in Fig. 3共a2兲. This result suggests that domain patterns

Downloaded 06 Jan 2009 to 117.32.153.167. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

262904-3

Appl. Phys. Lett. 93, 262904 共2008兲

J. Wang and M. Kamlah

domain structure can be realized by the specified nonuniform distributed surface charges shown in Fig. 3共c1兲. The four domains are separated by 180° domain walls. The domain walls are exactly located at the transition lines where the surface charges change their sign. In summary, the stable polarization distributions of ferroelectric nanodots with different surface charges are predicted by a phase field model. A critical charge density is found, at which the in-plane polarizations disappear and the ferroelectric dots experience a transition from a vortex state to a single-domain state. The out-of-plane polarization in the dots is found to be proportional to the density of surface charges. The simulations demonstrate that the desirable multidomain structures can be obtained by means of specific surface charges larger than the critical density. The ability to manipulate the domain patterns suggested in the present work holds promise as a method to tailor physical properties for the specific applications of ferroelectric nanostructures. J.W. gratefully acknowledges the Alexander von Humboldt Foundation for awarding a research fellowship to support his stay at Forschungszentrum Karlsruhe. FIG. 3. 共Color online兲 Domain controlling of ferroelectric nanodots by means of surface charges. 关共a1兲, 共b1兲, and 共c1兲兴 Three-dimensional spatial distribution of polarization component P3 in the ferroelectric nanodots. 关共a2兲, 共b2兲, and 共c2兲兴 Two-dimensional projection of the three-dimensional polarization vectors in the x2 − x3 plane at x1 = 8 nm. The single-domain 共a1兲, double-domain 共b1兲, and four-domain 共c1兲 structures are obtained, respectively, by specific surface charges.

in ferroelectric nanostructures can be manipulated by giving specific surface charges, which will be shown later. Figures 3共a1兲–3共c1兲 show the three-dimensional spatial distribution of polarization component P3 of the ferroelectric nanodots with three types of charge loadings. The arrows denote the orientation of the domains. The distributed positive and negative charges on surfaces are schematically shown by the circles with the sign ⫹ and ⫺, respectively. The normalized charge density is 1.0 for all the cases, which is equal to the critical value. Figures 3共a2兲–3共c2兲 give the detailed two-dimensional projection of the three-dimensional polarization vectors in the x2-x3 plane of the dots at x1 = 8 nm. When the uniform charges on the upper and lower surfaces are positive and negative, respectively, the polarizations in the ferroelectric dots form a single-domain structure as shown in Figs. 3共a1兲 and 3共a2兲. A double-domain structure is obtained when one-half of the surface charges change their sign as shown in Figs. 3共b1兲 and 3共b2兲. Here, the contour legend shows different values of polarization component P3 in different colors. There is a 180° domain wall between the two domains. Although the maximum polarization is larger than the polarization of a bulk single crystal and a single domain, the average value of 具兩P3兩典 is 0.68 C / m2. A four-

1

M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials 共Clarendon, Oxford, 1977兲. D. B. Li and D. A. Bonnell, Annu. Rev. Mater. Res. 38, 351 共2008兲. 3 J. Wang and M. Kamlah, Appl. Phys. Lett. 93, 042906 共2008兲. 4 I. Ponomareva, I. I. Naumov, and L. Bellaiche, Phys. Rev. B 72, 214118 共2005兲. 5 J. Wang, M. Kamlah, T. Y. Zhang, Y. L. Li, and L. Q. Chen, Appl. Phys. Lett. 92, 162905 共2008兲. 6 J. E. Spanier, A. M. Kolpak, J. J. Urban, L. Grinberg, L. Ouyang, W. S. Yun, A. M. Rappe, and H. Park, Nano Lett. 6, 735 共2006兲. 7 J. H. Ferris, D. B. Li, S. V. Kalinin, and D. A. Bonnell, Appl. Phys. Lett. 84, 774 共2004兲. 8 D. B. Li, M. H. Zhao, J. Garra, A. M. Kolpak, A. M. Rappe, D. A. Bonnell, and J. M. Vohs, Nature Mater. 7, 473 共2008兲. 9 R. Ahluwalia and W. Cao, Phys. Rev. B 63, 012103 共2000兲. 10 Y. L. Li, S. Y. Hu, Z. K. Liu, and L. Q. Chen, Appl. Phys. Lett. 78, 3878 共2001兲. 11 J. Wang, S. Q. Shi, L. Q. Chen, Y. L. Li, and T. Y. Zhang, Acta Mater. 52, 749 共2004兲. 12 W. Zhang and K. Bhattacharya, Acta Mater. 53, 185 共2005兲. 13 R. Muller, D. Gross, D. Schrade, and B. X. Xu, Int. J. Fract. 147, 173 共2007兲. 14 J. Slutsker, A. Artemev, and A. Roytburd, Phys. Rev. Lett. 100, 087602 共2008兲. 15 W. Cao and L. Cross, Phys. Rev. B 44, 5 共1991兲. 16 S. Nambu and D. Sagala, Phys. Rev. B 50, 5838 共1994兲. 17 Y. Su and C. M. Landis, J. Mech. Phys. Solids 55, 280 共2007兲. 18 For PbTiO3 共in SI units兲, a1 = 3.8共T − 479兲 ⫻ 105, a11 = 4.2⫻ 108, a12 = −2.68⫻ 108, a111 = 2.6⫻ 108, a123 = −3.7⫻ 109, c11 = 1.746⫻ 1011, c12 = 7.937⫻ 1010, c44 = 1.111⫻ 1011, q11 = 1.14⫻ 1010, q12 = 4.60⫻ 108, q44 = 7.49⫻ 109, g11 = 2.768⫻ 10−10, g12 = 0, and g44 = g⬘44 = 1.384⫻ 10−10. 19 N. Pertsev, A. Zembilgotov, and A. Tagantsev, Phys. Rev. Lett. 80, 1988 共1998兲. 20 L. Q. Chen, J. Am. Ceram. Soc. 91, 1835 共2008兲. 21 O. G. Vendik and S. P. Zubko, J. Appl. Phys. 88, 5343 共2000兲. 2

Downloaded 06 Jan 2009 to 117.32.153.167. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp