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Structure and energetics of 180° domain walls in PbTiO3 by density functional theory

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2011 J. Phys.: Condens. Matter 23 175902

(http://iopscience.iop.org/0953-8984/23/17/175902)

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J. Phys.: Condens. Matter 23 (2011) 175902 (12pp) doi:10.1088/0953-8984/23/17/175902

Structure and energetics of 180◦ domainwalls in PbTiO3 by density functionaltheoryRakesh K Behera1,5, Chan-Woo Lee1,6, Donghwa Lee1,7,Anna N Morozovska2, Susan B Sinnott1, Aravind Asthagiri3,8,Venkatraman Gopalan4 and Simon R Phillpot1,9

1 Department of Materials Science and Engineering, University of Florida, Gainesville,FL 32611, USA2 V Lashkaryov Institute for Semiconductor Physics, National Academy of Science ofUkraine, 41, prospekt Nauki, 03028, Kiev, Ukraine3 Department of Chemical Engineering, University of Florida, Gainesville, FL 32611, USA4 Department of Materials Science and Engineering, Pennsylvania State University,University Park, PA 16802, USA

E-mail: sphil@mse.ufl.edu

Received 14 October 2010, in final form 14 March 2011Published 14 April 2011Online at stacks.iop.org/JPhysCM/23/175902

AbstractDensity functional theory at the level of the local density approximation with the projectoraugmented wave method is used to determine the structure of 180◦ domain walls in tetragonalferroelectric PbTiO3. In agreement with previous studies, it is found that PbO-centered {100}walls have lower energies than TiO2-centered {100} walls, leading to a Peierls potential barrierfor wall motion along 〈010〉 of ∼36 mJ m−2. In addition to the Ising-like polarization along thetetragonal axis, it is found that near the domain wall, there is a small polarization in thewall-normal direction away from the domain wall. These Neel-like contributions to the domainwall are analyzed in terms of the Landau–Ginzburg–Devonshire phenomenological theory forferroelectrics. Similar characteristics are found for {110} domain walls, where OO-centeredwalls are energetically more favorable than the PbTiO-centered walls.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Domains and domain walls are central to applications offerroelectric materials ranging from nonvolatile random accessmemory, piezoelectric actuators and sensors, pyroelectricdetectors, and electro-optic and nonlinear optical devices [1].There have thus been considerable experimental [2–12],

5 Present address: Nuclear and Radiological Engineering Program,George W Woodruff School of Mechanical Engineering, Georgia Institute ofTechnology, Atlanta, GA 30332, USA.6 Present address: The Makineni Theoretical Laboratories, Department ofChemistry, University of Pennsylvania, Philadelphia, PA 19104, USA.7 Present address: Condensed Matter and Materials Division, LawrenceLivermore National Laboratory, Livermore, CA 94550, USA.8 Present address: Department of Chemical and Biomolecular Engineering,The Ohio State University, Columbus, OH 43210, USA.9 Author to whom any correspondence should be addressed.

theoretical [13–15], and computational [16–23] efforts tocharacterize domains in ferroelectric materials. PbTiO3 (PT)has the perovskite (ABO3) structure with Pb atoms occupyingthe cell corners, a Ti occupying the body center, and the oxygenatoms sitting at the face centers. The cubic paraelectric phasehas Pm3m symmetry, while the tetragonal ferroelectric phasehas P4mm symmetry. The cubic to tetragonal phase transitionat the Curie temperature typically leads to the developmentof 180◦ and 90◦ domains. The (100)-oriented 180◦ domainwalls could be centered at PbO planes (Pb-centered), TiO2

planes (Ti-centered), or between the two planes. The structureof such domain walls has previously been studied using firstprinciples calculations [16–20]. In particular, it was shownthat Pb-centered (100)-oriented 180◦ domain walls are morestable than the Ti-centered domain walls [16, 17]. However,most of the previous first principles studies only considered

0953-8984/11/175902+12$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA1

J. Phys.: Condens. Matter 23 (2011) 175902 R K Behera et al

Figure 1. Schematic of (a) a unit cell of PT, showing atomic displacements in the tetragonal ferroelectric phase, (b) a plane showing aPb-centered domain wall and (c) a plane indicating a Ti-centered domain wall. For (b) and (c), all the atoms in the domain wall are located intheir non-ferroelectric centrosymmetric positions.

atomic relaxation parallel to the direction of the uniaxial bulkpolarization. While at first glance this seems reasonable sincePT is an inherently uniaxial system (figure 1), the presence of adomain wall breaks the tetragonal symmetry thereby allowing,at least in principle, local atomic displacements in the twoCartesian directions perpendicular to the polarization. A recentwork by Lee et al [24] successfully characterized the presenceof in-plane polarizations in several ferroelectric materials(PbTiO3, LiNbO3, and BaTiO3). Following the work of Leeet al [24], this paper provides the detailed characterization ofPT domain walls using first principles calculations.

In this study, we use electronic-structure methods(section 2) to re-examine the structure of domain walls inwhich breaking of the uniaxial symmetry at the domain wallsis allowed. We find (section 3) that there is an additionalcomponent of polarization perpendicular to the domain wall.Section 4 discusses fitting this polarization to the Landau–Ginsburg–Devonshire (LGD) formalism to obtain the width ofthe domain wall. Section 5 summarizes our findings and offersconcluding remarks.

2. Methodology

All of the simulations are performed with density functionaltheory (DFT) at the level of the local density approximation(LDA) using the PAW (projector augmented wave) [25]method as incorporated in the Vienna ab initio simulationpackage (VASP) [26–28]. LDA typically underestimatesthe lattice parameters, whereas the generalized gradient ap-proximation (GGA) typically overestimates lattice parameters.Since we are interested in the polarization, which is verysensitive to the volume of the unit cell, we followed previouswork [16, 19, 29–31] by using the LDA. Such LDA calcula-tions yield bulk polarization values that are more consistentwith experiment than the GGA. We use PAW potentialsthat are currently state-of-the-art for DFT calculations; thesewere not available at the time of the earlier studies. Thepseudopotentials used for this study treat Pb 5d and Ti 3p statesexplicitly as valence states. All the calculations are performedwith 550 eV (∼40 Ryd) cutoff energies. Since the supercellsconsist of 1 × Y × 1 cubic perovskite unit cells, Y = 6–16;a 6 × 1 × 6 k-point Monkhorst–Pack [32] mesh is used forthe domain simulations. Convergence is taken to have beenachieved when the force on each atom reaches 0.01 eV A

−1.

A conjugate gradient method is used to optimize thedomain structures. The effect of the pseudopotentials,cutoff energy and optimization method on convergence hasbeen established based on previous studies for the PTsystems [19, 33, 34]. The zero stress lattice parameters forthe tetragonal phase of a = 3.8669 A and c/a = 1.0436,were obtained by optimizing the bulk tetragonal, ferroelectricstructure, and are used for all the subsequent DFT calculations.The experimental lattice parameters for the tetragonal phaseare a = 3.904 A and c/a = 1.063 [35]. The followingnotations are used interchangeably in the text: ferroelectricpolarization (Pz or P3) is along the Z -axis or [001] direction;the in-plane polarization (Pt or P1) is along the X -axis or [100]direction, and (Pn or P2) is along the Y -axis or [010] direction.

The (010)-oriented 180◦ domain walls can be centeredat either a Pb-centered or a Ti-centered plane (figure 1).Supercells of size 1 × Y × 1 with Y = 6, 8, 10, 12,14, and 16 are used for this study. In the initial structure,half of the PT unit cells are oriented with polarization inthe +z direction (up polarization), with the other half havingpolarization in the −z direction (down polarization). Sinceperiodic boundary conditions are applied in all three spatialdirections, this arrangement results in a pair of domain walls inthe x–z plane (figure 2 inset). Prior to structural equilibration,the atoms in the mirror plane of the domain walls themselvesare placed at their centrosymmetric positions.

For reasons that will become evident, we have used thefollowing structural optimization schemes:

(i) R–Z : the dimensions of the supercell are held rigid (R)

at values corresponding to the lattice constants calculatedfrom bulk PT. Atoms are only allowed to relax in the z-direction, i.e. parallel to the bulk uniaxial polarization,thereby maintaining the polarization symmetry of thesingle crystal.

(ii) R–XY Z : the supercell is held rigid with dimensions keptto the lattice constants calculated from bulk PT. Atoms areallowed to relax in all three spatial directions (XY Z ).

(iii) A–XY Z : the dimensions of the supercell are allowed toadjust to minimize the stress (A). Atoms can relax in allthree spatial directions.

For large enough separation between the domain walls,it is expected that the R–XY Z and A–XY Z results should

2

J. Phys.: Condens. Matter 23 (2011) 175902 R K Behera et al

Figure 2. Energies of (100) 180◦ domain wall energies for the R–Zand R–XY Z relaxation schemes for Y = 10: USPP refers to theresult published by Meyer and Vanderbilt using ultra-softpseudopotentials [16], R–Z and R–XY Z refers to the PAWcalculation results for the respective schemes. Inset is a schematic ofthe domain wall setup used in this study. There are two domainwalls, one at the center and the other at the end of the supercell. Thelattice parameters used to build the supercell are a = 3.8669 A andc = 4.0355 A.

converge, as we indeed find. However, there is a real physicaldifference between the Z and XY Z cases.

Finally, borrowing from the language of magnetism, thepolarization at the domain wall can be characterized in termsof Ising-like, Neel-like, and Bloch-like contributions. Figure 3illustrates the three kinds of domain walls. In general foruniaxial ferroelectrics, the domain wall is Ising type. However,the presence of non-uniaxial polarization at the domain wallcan develop a Bloch-like or Neel-like character [24]. Inparticular, polarization in the normal direction (Pn) indicatesa Neel-like rotation of the polarization vector (θN) in the n–zplane. The presence of polarization in the transverse direction(Pt) indicates a Bloch-like rotation (θB) in the t–z plane.

3. Domains in bulk PbTiO3

The tetragonal symmetry of the ferroelectric phase of PTrequires that atomic relaxation in the single crystal takes placealong only one of the Cartesian directions. However, thepresence of a domain wall breaks the tetragonal symmetrythereby, at least in principle, also allowing atomic relaxationin the two Cartesian directions orthogonal to the directionof polarization. All of the systems are analyzed in termsof the domain wall energy, atomic relaxations around thedomain wall, and components of polarization. The domainwall energies are defined in the usual manner as:

EDomain Wall = (EDomain − EBulk)

2A(1)

where EDomain is the energy of the system with the two domainwalls (hence the factor of two in the denominator), EBulk is theenergy of the equivalent system without the domain walls, andA is the area of the domain wall.

Figure 3. Schematic representation of (a) Ising type, (b) Bloch type,and (c) Neel type walls. A mixed wall will have contribution from acombination of two or all of the three kinds of walls. The Bloch andNeel rotations are represented by θB and θN, respectively.

3.1. Atomic relaxation in the z-direction only

Figure 2 shows the energies for Pb- and Ti-centered domainwalls calculated using the R–Z scheme for Y = 10. Meyerand Vanderbilt [16] previously used ultra-soft pseudopotentials(USPPs) to determine the structure of this domain wall. Wehave reproduced these results as a baseline against which tocompare our results. For both Pb- and Ti-centered walls, thePAW method gives slightly lower domain wall energies thanthe USPP calculations.

Both calculations predict that the Pb-centered (100) 180◦domain walls have lower energies (128 mJ m−2 in the PAWcalculation) than Ti-centered domain walls (163 mJ m−2).This energy difference of 35 mJ m−2 is very similar to thevalue of 37 mJ m−2 previously determined with the USPPcalculations [16]. Thus, following the work by Meyer andVanderbilt [16], the Ti-centered domain wall is the saddle-point configuration for the domain wall motion. Therefore, theminimum Peierls potential barrier for wall motion in the [010]direction is ∼35–37 mJ m−2.

The polarization was calculated for each supercell usingthe method of Meyer and Vanderbilt [16]. In this approach,the Born effective charges (Z∗

α) are calculated for eachcrystallographically distinct ion in the tetragonal unit cell usingPAW pseudopotentials (table 1). A comparison of Z∗

α valueswith previously reported Born effective charges from DFTcalculations [20, 29, 31] show excellent agreement. The i th(i = x , y, z) Cartesian component of the polarization of each

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J. Phys.: Condens. Matter 23 (2011) 175902 R K Behera et al

Table 1. Comparison of PAW and USPP calculated Born effective charges of the optimized structure for the cubic and tetragonal PbTiO3 (OI

and OIII are mentioned in figure 1).

Atom Z∗xx Z∗

yy Z∗zz

CubicPb 3.91 (3.90a) 3.91 3.91Ti 7.10 (7.06a) 7.10 7.10OI −2.60 (−2.56a) −5.84 (−5.83a) −2.60OIII −2.60 −2.60 −5.84TetragonalPb 3.84 (3.92a, 3.85b, 3.74c) 3.84 (3.85b, 3.74c) 3.49 (3.63b, 3.52c)

Ti 6.37 (6.71a, 6.37b, 6.20c) 6.37 (6.37b, 6.20c) 5.47 (5.41b, 5.18c)

OI −2.65(−2.56a,−2.66b,−2.61c) −5.26(−5.51a,−5.32b,−5.18c) −2.22(−2.22b,−2.16c)

OIII −2.24 (−2.24b,−2.15c) −2.24 (−2.24b,−2.15c) −4.64 (−4.60b, −4.38c)

a Vanderbilt ultra-soft pseudopotential (LDA)— [29].b Norm-conserving plane-wave pseudopotential (LDA)— [31].c Full potential ab initio LAPW method with local orbital extension (GGA)— [20].

unit cell is then calculated as

P(i) = e

�c

∑

α

wα Z∗αu(i)

α (2)

where e is the electron charge, �c is the volume of one unitcell, uα is the displacement of ion α from its centrosymmetricposition, and wα is the number of atoms of that type in the unitcell. In characterizing the polarization around a domain wall,it is also useful to define a contribution to the polarization ofeach individual ion:

P(i)α = e

�cZ∗

αu(i)α . (3)

Figure 4 shows the polarization profile determined fromequation (2) for supercells of various sizes (i.e. for varyingdistances between domain walls) for Pb-centered domainwalls. These results demonstrate that the polarization profile isessentially independent of the separation of the domains walls,except for the very smallest separation. Moreover, in everycase, the single-crystal bulk polarization (Pz = 86.7 μC cm−2)is recovered within only two lattice parameters from the centerof the domain wall. The polarization in the unit cell oneither side of the domain wall is ∼68.5% of the bulk value.The individual atomic relaxations determined with the PAWmethod are almost identical to those previously reported fromUSPP calculations [16]. The polarization profiles for the Ti-centered domain wall are similar to those of the Pb-centeredwall.

3.2. Atomic relaxation in all three spatial directions

To this point we have merely reproduced the results of previouscalculations on these domain walls. We next relax theconstraint on the displacement of atoms, allowing them tomove in all three spatial directions rather than in the z-directiononly.

Figure 2 also compares the domain wall energies for R–Z and R–XY Z relaxation. The removal of the constraint onthe ion displacements leads to a lowering of the energy by4.7 mJ m−2 (3.7 %) for Pb-centered and 3.4 mJ m−2 (2.1%)for Ti-centered walls compared to their respective Z -relaxationonly.

Figure 4. Cell-by-cell Z -polarization (Pz) variation of Pb-centereddomain systems with R–Z relaxation using PAW pseudopotential. Pz

is ∼68.5% around the domain wall for all the system sizes studiedand converges to the bulk value in the middle of each domain.Similar results are obtained for Ti-centered domain walls. Y = 06,08, and 10 indicates the total number of unit cells perpendicular tothe domain wall in the supercell.

Again we use the Meyer–Vanderbilt method to determinethe polarization (equation (2)). The polarization profile in thez-direction through the domain wall is essentially identical tothat found for the R–Z relaxation scheme. The ions do notmove parallel to the wall in the direction orthogonal to thepolarization; as a result there is no component of polarizationin this direction, Pt = 0. By contrast, however, there is now acomponent of polarization in the direction normal to the planeof the domain wall, Pn �= 0.

The presence of in-plane polarization has been previouslyreported in BaTiO3 [19], in which 180◦ domain walls intetragonal BaTiO3 were simulated using Monte Carlo methods,and the Ising-like nature of polarization along the tetragonalz-axis was identified. An additional polarization componentalong the parallel in-plane direction was observed; however,the spatial variations in these polarizations were reported to beuncorrelated with the domain wall position, and hence wereconsidered artifacts. The in-plane polarizations seen in thecurrent simulations are strongly correlated with the domainwall position and are not artifacts. This was evidenced by

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J. Phys.: Condens. Matter 23 (2011) 175902 R K Behera et al

Figure 5. Cell-by-cell Y -polarization (Pn) characterization forPb-centered domain systems with R–XY Z relaxation using PAWpseudopotentials. Pn is ∼2.5% of the bulk ferroelectric polarizationaround the domain wall for all the system sizes and the direction isaway from the domain wall. Y = 06–16 indicates the total number ofunit cells perpendicular to the domain wall in the supercell. The insetillustrates the development of polarization in the normal directionnear the domain wall (Pn).

separate simulations beginning from the R–Z configuration,in which the in-plane polarization emerged during further R–XY Z relaxation.

3.2.1. Pb-centered domain walls. As illustrated in figure 5,the polarization in the unit cell closest to the domain wall pointsaway from the domain wall. This polarization of 2.4 μC cm−2

is substantial, amounting to ∼2.5% of bulk Pz . Figure 5 alsoshows that the value of Pn in the unit cells proximate to thedomain wall is independent of the distance between the domainwalls. Interestingly, the Pn rapidly decays back to zero awayfrom the domain wall, indicating that there is no long-rangeeffect along the normal direction.

The corresponding atomic displacements in the directionnormal to the interface are calculated with respect to thepositions in the bulk ferroelectric, and evaluated as a functionof the distance to the domain wall. Figure 6 is a top-downview of a PbO plane and a TiO2 plane (the bulk polarizationis normal to these two planes). As figure 6 schematicallyindicates, all of the Ti ions and oxygen (OIII) ions on thePbO plane in the unit cells adjacent to the domain wall movetoward the domain wall. From equation (3) we can define aneffective contribution of each ion to the polarization. The Tiions move towards the domain wall by 0.01 A, correspondingto a polarization contribution of +1.25 μC cm−2 (i.e. towardsthe domain wall). The OIII ions relax the most (0.03 A)around the domain wall and contribute −3.61 μC cm−2 tothe polarization. The displacements of Pb ions, and theoxygen (OI and OII) ions are less than 0.005 A, and thereforedo not contribute significantly to the polarization. The netpolarization calculated from this Born effective charge analysisis thus 1.25−3.61 = −2.35 μC cm−2, which is consistent withthe magnitude and direction of Pn.

To this point we have presented results for the cases wherethe dimensions of the supercell are fixed to be the same as

Figure 6. Schematic of the atomic displacement of each type of ionaround the Pb-centered domain wall. The 〈001〉 view of (a) PbO and(b) TiO2 planes are shown with the arrows indicating the direction ofatomic movement. The oxygens on the PbO planes (OIII) move bythe largest amount (∼0.03 A), while the Ti atoms move ∼0.01 A.

the number of unit cells of a ferroelectric monodomain (R–Z and R–XY Z ). However, the optimization of the dimensionof the supercell also has an effect on the atomic relaxationand the polarization. For the Pb-centered domain wall withY = 16 using the A–XY Z scheme, we obtain a = 3.8731 Aand c/a = 1.0325, compared with the bulk values of a =3.8669 A and c/a = 1.0436. The polarization in the interiorof the domains only reaches 80.4 μC cm−2, compared withthe bulk value of 86.7 μC cm−2. The domain wall energy iscalculated to be 114 mJ m−2, which is somewhat lower than thevalue of 122 mJ m−2 determined for fixed lattice dimensions.Importantly, however, these calculations show a profile for Pn

that is essentially indistinguishable from that determined forfixed supercell dimensions.

3.2.2. Ti-centered domain walls. We have also performedthe corresponding analyses for the Ti-centered domain walls.Figure 7 represents the cell-by-cell normal polarization (Pn)for Y = 16. As in the case of the Pb-centered domain wall, thepolarization points away from the domain wall; the maximumvalue of Pn = −1.52 μC cm−2 (1.75% of bulk Pz) is obtainedone unit cell away from the domain wall.

Atomic displacements are calculated normal to the domainwall with respect to the positions in the bulk ferroelectric andevaluated as a function of the distance to the domain wall.Figure 8 shows a top-down view of a PbO and TiO2 plane(the bulk polarization is normal to these two planes). As wasthe case for the Pb-centered domain wall, all of the ions areobserved to move towards the domain wall. In particular,the oxygen (OIII) and Pb ions on the PbO plane in the unitcells near the domain wall move ∼0.023 A and 0.013 A,respectively. The displacements calculated for the rest of the

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J. Phys.: Condens. Matter 23 (2011) 175902 R K Behera et al

Figure 7. Cell-by-cell Y -polarization (Pn) for a Ti-centered domainwall system with R–XY Z relaxation using PAW pseudopotentials(Y = 16). The maximum value of Pn is ∼1.75% of the bulkferroelectric polarization, which is one lattice parameter away fromthe domain wall. The inset illustrates that the direction of Pn is awayfrom the domain wall. The polarization values at half the latticeparameter from the domain wall (indicated by open triangles) areignored for fitting purposes.

Figure 8. Schematic of the atomic displacement of each type of ionaround the Ti-centered domain wall. The 〈001〉 views of (a) PbO and(b) TiO2 planes are shown with the arrows indicating the direction ofatomic movement. The oxygens on the PbO planes (OIII) move bythe largest amount (∼0.023 A), while the Pb atoms move ∼0.013 A.The displacement of other atoms around the domain wall is observedto be <0.006 A, hence not shown in the schematic.

atoms (all the atoms on the TiO2 plane) are less than 0.006 A,and therefore do not contribute significantly.

3.2.3. (110) 180◦ domain walls. The domain calculationshave been extended to compare the structure and energeticsof (100) and (110) 180◦ domain walls. Structural analysis ofPT indicates two possible (110) 180◦ domain walls: PbTiO-centered and OO-centered (figure 9). The lattice parametersused for the (110) unit cells are a = 5.4687 A and c/a =

Figure 9. Schematic of a (110)-oriented PT unit cell, a = 5.4687 Aand c/a = 0.7379. The highlighted areas illustrate the location of apossible PbTiO-centered (the left area) and OO-centered (the rightarea) domain wall.

Figure 10. Cell-by-cell Z -polarization (Pz) variation for (100)Pb-centered and (110) OO-centered domain systems with R–XY Zrelaxation using the PAW pseudopotential. The variation in Pz

around the domain wall is similar for both the systems.

0.7379. All the calculations are performed with 1 × 6 × 1supercell size using PAW potential.

The domain wall energies calculated for R–Z schemesshow the OO-centered (110) 180◦ domain walls (124 mJ m−2)to have a lower energy than the PbTiO-centered domain wall(156 mJ m−2). We also compared the domain wall energies forR–Z and R–XY Z relaxation. The R–XY Z scheme leads to alowering of ∼3 mJ m−2 for both the (110) domain walls, whichis ∼2.3% for OO-centered and ∼1.9% for PbTiO-centeredwalls. We can thus estimate the minimum Peierls potentialbarrier for motion of the [110] walls as ∼32 mJ m−2, whichis comparable to the ∼35–37 mJ m−2 for the [010] direction.

Figure 10 gives the cell-by-cell polarization profile (Pz)for (100) Pb-centered and (110) OO-centered domain wallswith the R–XY Z optimization scheme. The variation inpolarization around the domain wall is similar for both cases,with the single-crystal bulk polarization (Pz = 86.7 μC cm−2)being recovered within only two bulk lattice parameters of PTfrom the center of the domain wall. The (110) OO-centereddomain walls also show a polarization perpendicular to the

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J. Phys.: Condens. Matter 23 (2011) 175902 R K Behera et al

Figure 11. Cell-by-cell normalized polarization fitting performed forPz (system size Y = 16) for (a) Pb-centered and (b) Ti-centereddomain walls. The open symbols are the data points obtained forpolarization in each unit cell from the optimized structure and thesolid lines are the fitting curves.

domain wall (Pn) with a maximum value of Pn ∼ 1.0 μC cm−2

(1.1% of bulk Pz) out of the domain wall. Overall, thecharacteristics of the (110) 180◦ domain walls are very similarwith those of the (100) 180◦ domain walls.

4. Discussion

4.1. Polarization analysis

The Landau–Ginsburg–Devonshire (LGD) formalism gives anexcellent description of the ferroelectric behavior of manymaterials. The symmetry invariant free energy terms for theparaelectric phase of PbTiO3 (m3m) can be written as a Taylorseries expansion of the Cartesian components (in the 〈100〉directions) of polarization (Pi (i = 1–3)) and stress (Xi ,i = 1–6 in Voigt notation) [36]. The LGD approach has beenwidely used to describe the bulk ferroelectric properties [36],ferroelectrics in the presence of surfaces [37], and ferroelectricnanostructures, such as ultrathin films [38], nanoparticles [39],and nanowires [40]. The LGD expressions used for fitting thedomain wall widths are summarized in the appendix.

4.1.1. (100) 180◦ domain walls. If only the Pz term isconsidered, then the one-dimensional LGD model yields adomain wall profile given by [41]:

Pz(x) = PS tanh ((x − x0)/2L⊥)√η sech2 ((x − x0)/2L⊥) + 1

(4)

where PS is the bulk polarization, x is the distancefrom the center of the domain wall at x0, and 2L⊥ isthe domain wall width. For second-order ferroelectricsP2

S = −a3/a33 and η = 0, while for first-order

ferroelectrics P2S = (

√a2

33 − 4a3a111 − a33)/2a111 and

η = 2(a3 + a33 P2S )/(4a3 + a33 P2

S ). The correlation length

is L⊥ =√

g12/(a3 + 3a33 P2S + 5a111 P4

S ). For η = 0the solution of equation (4) reduces to the well-known one-dimensional expression for the domain wall profile [41]. Thedomain wall profile in figure 4 for the R–Z relaxation casefor (100) domain walls can be well-fitted to equation (4) withPS = 86.7 μC cm−2 (the bulk polarization value) and 2L⊥ =2.12 A for Pb-centered domain walls and 2L⊥ = 2.66 A forTi-centered domain walls.

The approximate analytical solution for the mixed domainwalls with Pz(x) �= 0 and Pn,t(x) �= 0 can be found asa combination of soliton-like and kink-like exact solutions.Considering the odd functions, which fit to the polarizationprofile, the mixed Ising–Neel type solutions are given as:

Pz (x) ≈ PS tanh (qn (x − x0))√η sech2 (qn (x − x0)) + 1

, (5a)

Poddn (x) ≈ Qodd

1 tanh (qn (x − x0))

cosh (qn (x − x0)), Pt (x) ≈ 0 (5b)

where the wavevector qn is represented as

qn =√

1

g11

(nd

ε0+ a1 + a13 P2

S + a112

2P4

S

)(6)

and the odd Ising–Neel type amplitude is given by

Qodd1 =

(−6b2 − 4a23 P4S − a112 P4

S

2(a22 + a112 P2S )

+{(

6b2 + 4a13 P4S + a112 P4

S

2(a22 + a112 P2S )

)2

+ 2b2 P2S + P4

S (2a23 − a33 − (2a111 − a112)P2S )

a22 + a112 P2S

}1/2)1/2

(7)

where b2 = a1 + (nd/ε0).All of the polarization profiles from the DFT calculations

are fitted to equations (5a) and (5b), thereby allowing usto determine the domain wall widths for the mixed Ising–Neel type walls. Figure 11 shows the normalized polarization(Pz/Ps ) fitting profiles for both Pb-centered and Ti-centereddomain walls. For both the domain walls, the widths calculatedfrom fitting equation (5a) (qn) are the same as those calculatedfrom equation (4) (2L⊥). Similarly, the in-plane polarizationprofiles are fitted to equation (5b) (figure 12). The Pn

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J. Phys.: Condens. Matter 23 (2011) 175902 R K Behera et al

Table 2. Calculation of g11 and g12 to develop a consistent set of fitting data for qn, Qodd1 , and nd values using the analytical expression and

fitting polarization profiles obtained for (100) 180◦ domain walls in PbTiO3.

Type of the domain wall 2L⊥ (A) qn (A−1

) Qodd1 (C m−2) nd g11 (C−2 m4 N) g12 (C−2 m4 N) g11/g12

Pb-centered 2.12 0.523 1/qn = 1.91 A 0.044 0.009 220 4.55 × 10−11 5.59 × 10−11 0.81Ti-centered 2.66 0.314 1/qn = 3.18 A 0.033 0.009 201 1.26 × 10−10 8.79 × 10−11 1.43

Figure 12. Cell-by-cell polarization fitting performed for Pn (systemsize Y = 16) for (a) Pb-centered and (b) Ti-centered domain walls.The open symbols are the data points obtained for polarization ineach unit cell from the optimized structure and the solid lines are thefitting curves. The polarization values at half the lattice parameterfrom the domain wall are ignored for the Ti-centered domain wall.

polarization values are well described around the domainwall. The polarization profiles for Pb-centered domain walls(figure 12(a)) are well-fitted with equation (5b). However, forTi-centered walls, the LGD fit only qualitatively described theDFT calculated Pn values (figure 12(b)) near the immediatevicinity of the domain wall. These differences do not invalidatethe analysis since the magnitudes of these polarizations arevery small (less than 0.5 μC cm−2). The domain wall widths(1/qn) predicted using equation (5b) are 1.91 A for Pb-centeredand 3.18 A for Ti-centered domain walls. This is consistentwith the observation of the atomic displacements shown infigures 6 and 8, where the Pb-centered domain walls show in-plane atomic displacements only in the adjacent atomic planesto the wall, while the Ti-centered domain walls affect a unitcell around the wall.

This analysis also allows us to estimate values of a numberof parameters that are not experimentally well known: L⊥, qn,Qodd

1 , nd, g11, and g12. It is important to point out that thevalues of the following coefficients from equation (A.1), αi

(including the Curie temperature 752 K), αi j , αi jk , si j (elasticmodules) and Qi jk (electrostriction coefficients), are well-established materials’ constants for PbTiO3, obtained fromvarious experimental results (listed in table A.1). Thus, thepolarization profiles predicted with DFT for the domain wallsare fitted by a total of only three unknown coefficients (nd, g11,and g12) to provide a consistent data set. Table 2 summarizesthe values of each individual variable tabulated for both Pb-and Ti-centered domain walls. The gradient coefficients (g11

and g12) are average values and can be expected to depend onthe local structure around the defects or domain walls [42–44].Therefore, g11 and g12 are calculated to make a consistent setof data points to match the fitting profiles for both the (100)180◦ domain walls. The values of g11 and g12 are in therange of 10−10–10−11 C−2 m4 N, and the g11/g12 are calculatedto be 0.81 and 1.43 for Pb- and Ti-centered domain walls,respectively.

5. Closing remarks

Classic antiparallel (180◦) ferroelectric walls are well knownto be Ising type. However, here we have shown that Blochand Neel-like components can exist at such walls in PbTiO3.In particular, the structure and energetics of 180◦ domainwalls in PbTiO3 have been characterized with a first principlesapproach (PAW pseudopotentials). It was found that alongwith the primary polarization (Pz ), there is an additional wall-normal component to the polarization (Pn) for both (100)and (110) 180◦ domain walls in PTiO3. The presence ofthe in-plane polarization develops a mixed Ising–Neel typedomain wall [24]. A corresponding study on domain walls inLiNbO3 also suggested the presence of in-plane polarization,indicating the domain walls in ferroelectric materials as mixedtypes (Ising–Neel or Ising–Bloch) rather than pure Isingtype [24, 45]. An LGD analysis, extended to include non-uniaxial components of polarization, was shown to yield Ising–Neel type domain walls comparable to those observed in theDFT calculations. The Neel rotation (θN) was characterized tobe less than 2◦ for all the domain walls in PbTiO3.

For 180◦ domains, (100) Pb-centered and (110) OO-centered walls were found to be energetically more favorablecompared to (100) Ti-centered and (110) PbTiO-centeredwalls. Thus, based on the energetics, motion of 180◦ domainwalls is expected to show a combination of both (100) and(110) terminations. In fact, the results published by Shin et al[14] for the nucleation and growth of PbTiO3 through a diffuse

8

J. Phys.: Condens. Matter 23 (2011) 175902 R K Behera et al

boundary model shows a combination of (100) Pb-centeredwalls and walls slanted 45◦ relative to the lattice vectors of thePbTiO3 unit cell. From our calculations, we can predict thatthe combination of the planes for the nucleation and growthpresented by Shin et al [14] will be (100) Pb-centered and(110) OO-centered walls.

The presence of additional in-plane polarization can beimportant in establishing the observed macroscopic propertiesof ferroelectric materials. For example, the threshold fieldfor wall motion is exponentially dependent on the widthof the domain wall [46]. The Neel component can resultin domain wall widths which are twice that due to theuniaxial polarization only, therefore significantly influencingthe domain wall motion.

The influence of defects on the domain wall dynamics isanother important issue. The study by He and Vanderbilt [47]reported the influence of an oxygen-vacancy on 180◦ domainwalls in PbTiO3. The results showed a tendency of the oxygenvacancies to migrate to the domain wall and pin the motion.It will be of significant interest to characterize the mutualinteraction of point defects with the Neel component of thepolarization.

Acknowledgments

We thank the National Science Foundation contracts DMR-0426870, DMR-0820404, 0602986, 0512165, and 0507146.We also thank the High Performance Computing (HPC),University of Florida for providing resources for the densityfunctional theory calculations. The authors gratefullyacknowledge Dr E A Eliseev and Professor L Q Chen forvaluable discussions and critical remarks.

Appendix

The symmetry invariant free energy density, Gb, for theparaelectric phase of PbTiO3 (m3m) can be written as a Taylorseries expansion of the Cartesian components (in the 〈100〉directions) of polarization, Pi (i = 1–3), and of the stress Xi

(i = 1–6 in Voigt notation) as [36]:

Gb(x, y, z) =(

αX1

(P2

1 + P22 + P2

3

) + αX11

(P4

1 + P42 + P4

3

)

+ αX12

(P2

1 P22 + P2

2 P23 + P2

3 P21

) + αX111

(P6

1 + P62 + P6

3

)

+ αX112[P4

1 (P22 + P2

3 ) + P42 (P2

3 + P21 )

+ P43 (P2

1 + P22 )] + αX

123

(P2

1 P22 P2

3

)

− 12 s P

11

(X2

1 + X22 + X2

3

) − s P12 (X1 X2 + X2 X3 + X3 X1)

− 12 s P

44(X24 + X2

5 + X26) − Q11(X1 P2

1 + X2 P22 + X3 P2

3 )

− Q12[X1(P22 + P2

3 ) + X2(P23 + P2

1 ) + X3(P21 + P2

2 )]− Q44 (X4 P2 P3 + X5 P3 P1 + X6 P1 P2)

− PiEd

i

2+ gi j

2

(∂ Pi

∂x j

)2). (A.1)

Gradients coefficients gi j are regarded positive forcommensurate ferroelectrics. Here αX

i , αXi j , and αX

i jk arethe dielectric stiffness and higher-order stiffness coefficientsat constant stress; s P

i j is the elastic compliance coefficient

at constant polarization; Qi j is the electrostriction tensor(table A.1). The second-order terms in the free energy functiondo not have a preferential direction of polarization. The fourth-order terms have minima along the 〈100〉 axes. The sixth-order terms have minima along the 〈110〉 direction. Hence,lowering the temperature will favor a change in the direction ofpolarization from [001]-oriented to [011]-oriented. Hereinafterwe refer to P1 and P2 as the in-plane polarizations (Neel andBloch components), and P3 = Pz (ferroelectric polarization).

Edi is the depolarization field, resulting from imperfect

screening by the surrounding inhomogeneous polarizationdistribution and/or its discontinuities at interfaces. Thedistribution Pi(x, y, z) may induce inhomogeneous stress. Toyield the equilibrium stress distribution, the free energy inequation (A.1) should be minimized with respect to stresstensor components, X jk , as ∂G/∂ X jk = −u jk , whereui j is the strain tensor. Additional mechanical equilibriumconditions ∂ Xi j(x)/∂xi = 0 and the compatibility relation,eikl e jmn(∂

2uln/∂xk∂xm) = 0 (eikl is the permutation symbolor anti-symmetric Levi-Civita tensor), as well as mechanicalboundary conditions for zero stress at mechanically freesurfaces should be satisfied. For the cases of the clampedsystem with defined displacement components (or with mixedboundary conditions) one should find the equilibrium stateas the minimum of the Helmholtz free energy F = G +∫

V d3 ru jk X jk originating from Legendre transformation of G.After the mechanical equilibrium constraints are imposed,

the LGD free energy bulk density is:

Gb =(

a1

2P2

1 + a2

2P2

2 + a3

2P2

3 + a12

2P2

1

(P2

2 + P23

)

+ a23

2P2

3 P22 + a11

4P4

1 + a22

4

(P4

2 + P43

)

+ a111

6

(P6

1 + P62 + P6

3

) + a123

2P2

3 P22 P2

1

+ a112

4(P4

3 (P21 + P2

2 ) + P41 (P2

2 + P23 ) + P4

2 (P21 + P2

3 ))

+ 12

(g11 (∇ P1)

2 + g12((∇ P2)

2 + (∇ P3)2))

− 12 (P3 Ed

3 + P2 Ed2 + P1 Ed

1)

)(A.2)

where ∇ is the gradient operator and Edi is the depolarization

field component. For ferroelectrics–dielectrics, the depolariza-tion field components Ed

i = −∂ϕd/∂xi and can be found fromthe electrostatic equation:

εb11

(∂2ϕd

∂x2+ ∂2ϕd

∂y2

)+ εb

33

∂2ϕd

∂z2= 1

ε0

(∂ P1

∂x+ ∂ P2

∂y+ ∂ P3

∂z

)

(A.3)where εb

ii are the background permittivity tensor componentsand ε0 = 8.85 × 10−12 F m−1 is the universal dielectricconstant. Corresponding boundary conditions determine thepossible screening of depolarization field.

The renormalized coefficients in equation (A.2) are:

a1 = 2

(αX

1 − Q12(Q11 + Q12)

s11 + s12P2

S

),

a2 = 2

(αX

1 +(Q2

11 + Q212

)s12 − 2s11 Q11 Q12

s211 − s2

12

P2S

),

(A.4a)

9

J. Phys.: Condens. Matter 23 (2011) 175902 R K Behera et al

Table A.1. Free energy parameters for bulk PbTiO3 [36, 49]. (Note: the unit F ≡ C2 m−1 N−1.)

Landau free energyparameters Value

Elastic energyparameters Value

α1 (105 C−2 m6 N) 3.8 (T -752 K) s11 (10−12 m2 N−1) 8.0α11 (108 C−4 m6 N) −0.73 s12 (10−12 m2 N−1) −2.5α12 (108 C−4 m6 N) 7.5 s44 (10−12 m2 N−1) 9.0α111 (108 C−6 m10 N) 2.6 Gradient energy parameters Valueα112 (108 C−6 m10 N) 6.1 g11 = G11/G110 (m3 F−1) 2.0 × 10−10

α123 (108 C−6 m10 N) −37 g12 = G12/G110 (m3 F−1) 1.0 × 10−10

Q11 (C−2 m4) 0.08 g44 = g12 (m3 F−1)Q12 (C−2 m4) −0.026Q44 (C−2 m4) 0.0337

a3 = 2

(αX

1 − (Q211 + Q2

12)s11 − 2s12 Q11 Q12

s211 − s2

12

P2S

), (A.4b)

a11 = 4αX11 + 4

Q212

s11 + s12,

a22 = a33 = 4αX11 + 2

(Q2

11 + Q212

)s11 − 2s12 Q11 Q12

s211 − s2

12

,

(A.4c)

a12 = a13 = 2αX12 + 2

Q12 (Q11 + Q12)

s11 + s12,

a23 = 2αX12 − 2

(Q2

11 + Q212

)s12 − 2s11 Q11 Q12

s211 − s2

12

+ Q244

s44.

(A.4d)a111 = 6αX

111, a112 = 4αX112, a123 = 2αX

123.

(A.4e)Variation of the LGD free energy functional (equa-

tion (A.2)) leads to coupled nonlinear Euler–Lagrange equa-tions

(a3 + a23 P2

2 + a13 P21

)P+

3 a33 P33 + f3 − g12P3 = Ed

3 ,

(A.5a)(a2 + a23 P2

3 + a12 P21

)P+

2 a22 P32 + f2 − g12P2 = Ed

2 ,

(A.5b)(a1 + a13 P2

3 + a12 P22

)P+

1 a11 P31 + f1 − g11P1 = Ed

1 .

(A.5c)The functions fi = a111 P5

i +a123 Pi P2j P2

k +a112(P3i (P2

j + P2k )

+ Pi (P4j + P4

k )/2) depend on the fifth powers of polarization(i , j , k = 1, 2, 3, i �= j �= k) and thus can be neglected forthe ferroelectrics that undergo a second-order phase transition. = ∂2

∂x2 + ∂2

∂ y2 + ∂2

∂z2 is the Laplace operator.To enable further analytical analysis, we assume that

|P1(x)| � |P3,2(x)| and neglect P1(x) in equations (A.5a)and (A.5b) in the actual range of parameters. Thus,(

a3 + a23 P22 + a112

2P4

2

)P3 + (

a33 + a112 P22

)P3

3

+ a111 P53 − g12

d2 P3

dx2= 0, (A.6a)

(a2 + a23 P2

3 + a112

2P4

3

)P+

2

(a22 + a112 P2

3

)P3

2

+ a111 P52 − g12

d2 P2

dx2= 0, (A.6b)

P1

(nd

ε0+ a1 + a12 P2

2 + a13 P23 + a112

2

(P4

2 + P43

))

+ (a11 + a112

(P2

3 + P22

))P3

1 + a111 P51 − g11

d2 P1

dx2= 0.

(A.6c)

Note that the term nd∼= Nd

εb11

in equation (A.6c) originated from

the depolarization field x-component Ed1(x) and its effective

screening. Ed1 (x) ∼ −Nd

P1(x)

ε0εb11

is strong enough for the ‘bare’

case, where Nd = 1 since ε0 is a large number. In a morefavorable situation, the bare field could be locally screened byappropriate choice of the boundary conditions [48] or internalscreening mechanisms and one may introduce the effectivedepolarization factor 0 < Nd � 1.

In the absence of P1,2(x), the solution of equation (A.6a)can be given in the form of a well-known single Ising typedomain wall as:

P3(x) = PS tanh ((x − x0)/2L⊥)√η sech2 ((x − x0)/2L⊥) + 1

. (A.7)

For second-order ferroelectrics P2S = −a3/a33 and

η = 0, while for first-order ferroelectrics P2S =

(

√a2

33 − 4a3a111 − a33)/2a111 and η = 2(a3 + a33 P2S )/(4a3+

a33 P2S ). The correlation length is

L⊥ =√

g12/(a3 + 3a33 P2S + 5a111 P4

S ), where x = x0

represents the domain wall plane. However, the simplersolution for the materials with a second-order phase transition(a33 > 0) can be used as an approximation for the first-orderones (a33 < 0, a111 > 0).

Let us look for the non-trivial mixed Ising–Bloch typeand Ising–Neel type solutions of equations (A.6) under theassumption that |P3,2| |P1|. The approximate analyticalsolution for the mixed Ising–Bloch and Ising–Neel typedomain walls with P3(x) �= 0 and P2,1(x) �= 0 can be found asa combination of soliton-like and kink-like exact solutions ofthe one-parameter equations, e.g. as odd functions

P3(ξ) ≈ PS tanh (qiξ)√η sech2 (qiξ) + 1

(1 + δpi

coshn (qiξ)+ · · ·

),

Pi = Qisinhl (qiξ)

coshm (qiξ)

10

J. Phys.: Condens. Matter 23 (2011) 175902 R K Behera et al

or even functions

Pi = Qi

coshm (qiξ)(where ξ = x − x0, i = 1, 2).

The deviation in Pi vanishes for natural numbers n � 1, l � 1and m > l at |ξ | → ∞.

The odd and even Ising–Bloch type solutions are derivedas:

P3 (x) ≈ PS tanh (qB (x − x0))√η sech2 (qB (x − x0)) + 1

, (A.8a)

Podd2 (x) ≈ Qodd

2 tanh (qB (x − x0))

cosh (qB (x − x0)),

Peven2 (x) ≈ Qeven

2

cosh (qB (x − x0)), P1 (x) ≈ 0.

(A.8b)

where the wavevectors are represented as:

qN =√

1

g11

(nd

ε0+ a1 + a13 P2

S + a112

2P4

S

),

qB =√

1

g12

(a2 + a23 P2

S + a112

2P4

S

).

(A.9)

The odd and even Ising–Bloch type amplitudes are:

Qodd2 =

(−6a2 − 4a23 P4S − a112 P4

S

2(a22 + a112 P2

S

)

+{(

6a2 + 4a13 P4S + a112 P4

S

2(a22 + a112 P2

S

))2

+ 2a2 P2S + P4

S (2a23−a33−(2a111−a112)P2S )

a22 + a112 P2S

}1/2)1/2

,

(A.10a)

Qeven2 =

( −2a2 + a112 P4S

2(a22 + a112 P2

S

) +{(−2a2 + a112 P4

S

2a22 + 2a112 P2S

)2

+ 2a2 P2S + P4

S (2a23−a33−(2a111−a112)P2S )

a22 + a112 P2S

}1/2)1/2

.

(A.10b)

Similarly, the odd and even Ising–Neel type solutions are:

P3 (x) ≈ PS tanh (qN (x − x0))√η sech2 (qN (x − x0)) + 1

, (A.11a)

Podd1 (x) ≈ Qodd

1 tanh (qN (x − x0))

cosh (qN (x − x0)),

Peven1 (x) ≈ Qeven

1

cosh (qN (x − x0)), P2 (x) ≈ 0

(A.11b)

where the odd and even Ising–Neel type amplitudes could beobtained from equations (A.10a) and (A.10b), respectively, bysubstituting a2 → a1 + (nd/ε0).

The approximate analytical LGD results equations (A.8)–(A.11) demonstrate the possibility of both Ising–Bloch andIsing–Neel type domain wall appearance at reasonable materialconstants and small screening factor. The choice between theodd end even solutions should be made only after comparisonof corresponding LGD free energies.

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