Journal of the Korean Physical Society, Vol. 52, No. 4, April 2008, pp. 1206�1210

Bond-Valence Model of Ferroelectric PbTiO3Young-Han Shin and Byeong-Joo LeeDepartment of Materials Science and Engineering,Pohang University of Science and Technology, Pohang 790-784

Andrew M. Rappe�The Makineni Theoretical Laboratories, Department of Chemistry,University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323, U.S.A.(Received 12 November 2007)A classical potential parameterized for the reproduction of density functional calculations is usedto describe the behavior of complex ferroelectric PbTiO3. A scoring function is de�ned in termsof the energy di�erences and the forces of the reference structures that are generated from ab-initio molecular dynamics simulations with various strained lattice vectors. The elastic propertiesof ferroelectric PbTiO3, as well as the phase transition temperature, have been improved by theaddition of the strained reference structures.PACS numbers: 77.80.-e, 02.70.NsKeywords: Ferroelectric, Density functional theory, Bond-valence modelI. INTRODUCTION

Ferroelectricity in perovskites has prompted wide in-vestigation due to applications, such as nonvolatile ran-dom access memories [1{7]. When an external electric�eld is applied to ferroelectric materials, the polariza-tion direction changes along the �eld direction and thispolarization state remains after the �eld is turned o�.In studies of the �eld dependence of ferroelectrics byMerz, Miller and Weinreich and Stadler [1,8,9], the nucle-ation rates were found to have an exponential relation tothe external electric �elds, which is known as Merz law.Tybell et al. recently observed Merz law in PZT thin�lms by using atomic force microscopy and showed thatthe size of each ipped domain could be controlled tomake high-density nonvolatile random access memories[3]. However, a detailed experimental understanding ofthe switching process is still hampered by the di�culty indetecting the fast domain wall motion and separating theintrinsic domain wall speed from extrinsic e�ects. Eventhough density-functional theory (DFT) calculations arerequired for this study, they demand too many computa-tions to observe the dynamical property of ferroelectrics.Atomic potential models have been designed to clarifythe physical phenomena that are hidden in real complexsystems. These gave opportunities to approach large sys-tems, like a moving domain wall, with available computerresources. The �rst model for an ionic system was therigid ion model, sometimes called the Born model. In�E-mail: rappe@sas.upenn.edu

the rigid ion model, the ions interact with each otherthrough the Coulomb interaction, but the polarizabilityof atoms is not considered in this model. Following therigid ion model, the shell model was �rst developed byDick and Overhauser to include the e�ect of the polar-izability [10,11]. A massless spherical shell of charge �qsurrounds an ion of charge (Z+q) and is attached to theion by a harmonic spring. In a modi�ed version of theshell model, the radius of the shell is a variable.Recently, another potential model has been suggestedfrom a completely di�erent idea { the inverse relation be-tween the bond valence (or the bond order) and the bondlength. This empirical relation was developed from an in-organic experimental database and its parameters weresummarized by Brown et al. [12{14]. The bond-valencemodel was successfully applied to PbTiO3 [15{17]. Here,we are going to show how the potential parameters areoptimized from the �rst-principles reference structures.We added more reference structures to consider the elas-tic properties, which enabled us to run molecular dynam-ics simulations at a constant stress.II. BOND-VALENCE MODEL ANDOPTIMIZATION OF PARAMETERS

The bond-valence model is based on the empiricaldatabase relating the bond lengths to the bond valences[12]. We create an interatomic potential from bond-valence concepts by adding several energy terms:Ebv = Ec + Er + Eb + Ea: (1)-1206-

Bond-Valence Model of Ferroelectric PbTiO3 { Young-Han Shin et al. -1207-Table 1. Optimized potential parameters of the bond-valence model potential function. The angle potential pa-rameter k is 1.43 meV/(�)2.

B��0 (�A)� r0�O C�O q� (e) S� (eV) Pb Ti OPb 1.969 5.5 1.419 0.013 - 2.224 1.686Ti 1.804 5.2 1.036 0.223 2.224 - 1.201O - - {0.818 0.702 1.686 1.201 1.857Ec is a Coulombic potential that is similar to the onein the Born model with charges of ions and these par-tial charges are �tting parameters. Er is a short-rangepotential with only a repulsive term:

Er = 8" 3X�=1

N�Xi=1

3X�0=1

N�0Xi0=1

B��0r��0ii0!12 : (2)

The relation between the bond length and the bond va-lence is expressed with Eb asEb = 3X

�=1S�N�Xi=1 (Vi� � V�)2 ; (3)

whereVi� = 3X

�0=1X

i0(n:n:) r��0or��0ii0

!C��0 (4)is the bond valence of the ith atom of type �. � is theindex for Pb, Ti and O ions and N� is the number of �ions. V� is the desired atomic valence of the � ion andr��0ii0 is the distance between the ith � ion and the i0th�0 ion. The angle potential Ea is introduced to correctfor the octahedral tilt at high temperatures.Reference structures are required to optimize the po-tential parameters. We generate the reference structuresby using ab-initio molecular dynamics simulations as im-plemented in VASP [18,19]. The local density approxi-mation was used for exchange and correlation and theprojector augmented wave potential was used for the

pseudopotential [20]. The plane waves were included inthe wave functions up to 500 eV. A 2 � 2 � 2 supercellwas used to consider the octahedral tilt angle and 14 kpoints were used for the k space integration in the irre-ducible Brillouin zone. The optimized cell parameters ofthe tetragonal PbTiO3 are a = 3:87 �A and c = 4:05 �A. Inaddition to the ab-initio molecular dynamics simulations,we performed static ab-initio calculations of the relaxedatomic coordinates with a series of strained lattice vec-tors. This served as a way to check whether this informa-tion would be helpful in adjusting the elastic properties.We used the simulated annealing algorithm to �nd theoptimized potential parameters. The scoring function,P , is de�ned asP = 1Ns

NsXi=1 (wE�Ei + wF�Fi) ; (5)

where�Ei = ��(EiDFT � E0DFT )� (Eibv � E0bv)�� ; (6)�Fi = NaX

j=13X

m=1���F i;j;mDFT � F i;j;mbv ��� : (7)

Ns is the number of reference structures and Na is thenumber of atoms. E0DFT and E0bv are the minimal DFTand the model potential energies among all the referencestructures, respectively. The upper indices i, j and m,respectively, denote the reference structures, atoms anddimensions. wE is the weight for the energy term andwF is the weight for the force term. These weights mustbe determined carefully to achieve optimal parameters.Since the lattice constants are underestimated in thelocal density approximation, Brown's empirical parame-ters r��00 and C��0 need to be adjusted. This is usefulfor generating the potential parameters of other ferro-electrics, where the empirical parameters are not known.We �x C��0 to the empirical values (CPbO = 5.5 and CTiO= 5.2) and express r��00 in terms of the displacements ofions (�Pb, �Ti, �O and �O0) and the lattice constants aand c (Figure 1):

rPbO0 =24 1=2�(a2 )2 + ( c2 � �Pb � �O)2��CPbO2 + �(a2 )2 + ( c2 + �Pb + �O)2��CPbO2 + �2(a2 )2 + (�Pb + �O0)2��CPbO2

351=NPbO

= 1:969 (8)rTiO0 =

24 4� c2 � �Ti � �O0��CTiO2 + 4 �(a2 )2 + (�Ti + �O)2��CTiO2 + � c2 � �Ti � �O0��CTiO2

351=NTiO = 1:804; (9)

where �Pb = 0:234 �A, �Ti = 0:096 �A, �O = �0:188 �A and�O0 = �0:137 �A from the �rst-principles calculation. The calculated r��00 parameters are slightly smaller than the

-1208- Journal of the Korean Physical Society, Vol. 52, No. 4, April 2008

Fig. 1. Nearest-neighbor oxygen ions around Pb and Tiions. (a) A solid circle designates a Pb ion and 12 oxygen ions(open circles) are located at the same distance from the Pbion. Arrows show the displacement of ions in the ferroelectricstate. (b) A solid circle designates a Ti ion and 6 oxygen ions(open circles) are located at the same distance from the Tiion.experimental values (rPbO0 = 2:044 and rTiO0 = 1.806).The optimized potential parameters are summarized inTable 1 and the energy di�erence between DFT and thebond-valence model is shown in Figure 2. The averageof the energy di�erence is 0.0095 eV per atom with astandard deviation of 0.0057 eV per atom.III. ELASTICITY OF TETRAGONAL PbTiO3The elastic energy per unit volume is given byE = 12Xi;j Cij�i�j : (10)

Cij is an element of the 6�6 symmetric sti�ness tensor Cand �i is a strain component following the Voigt notation[21]:0B@ �1 �6 �5�6 �2 �4�5 �4 �31CA =

0B@ �11 2�12 2�132�12 �22 2�232�13 2�23 �331CA : (11)

Fig. 2. Comparison of the energy di�erence between theDFT and the model potential. Around 3200 reference struc-tures with strained lattice vectors were used to optimize thepotential parameters.��� is de�ned as

��� = 12�@u�@x� + @u�@x�

� ; (12)where u is a displacement vector. By introducing a ma-trix T and , we can write � as

� = T: (13)For a tetragonal structure, the sti�ness tensor C is assimple as

C =0BBBBBBB@

C11 C12 C13 0 0 0C12 C11 C13 0 0 0C13 C13 C33 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 C66

1CCCCCCCA; (14)

and the elastic energy isE = 12C11 ��21 + �22�+ 12C33�23 + 12C44 ��24 + �25�

+12C66�26 + C12�1�2 + C13 (�1�3 + �2�3)= 12C11 ��211 + �222�+ 12C33�233+2C44 ��223 + �213�+ 2C66�212 + C12�11�22+C13 (�11�33 + �22�33) : (15)

The elements of the elastic sti�ness tensor are obtainedby determining proper T tensors (see Table 2) [22] andthey show good agreement with experiments [23,24].When the model potential energies with the originalparameter set in Ref. 14 is calculated with the six straintensors in Table 2, the energy di�erences between DFTand the model calculations for the T(11), T(13) and

Bond-Valence Model of Ferroelectric PbTiO3 { Young-Han Shin et al. -1209-Table 2. Strain tensors, elastic energies and the elastic sti�ness tensor C.

T � E Elements of the sti�ness tensorT(11)=

0B@

1 0 00 0 00 0 01CA

0B@

0 00 0 00 0 01CA 12C11 2 C11 = @2E@ 2

T(33)=0B@

0 0 00 0 00 0 11CA

0B@

0 0 00 0 00 0 1CA 12C33 2 C33 = @2E@ 2

T(44)=0B@

0 0 10 0 01 0 01CA

0B@

0 0 0 0 0 0 01CA 2C44 2 C44 = 14 @2E@ 2

T(66)=0B@

0 1 01 0 00 0 01CA

0B@

0 0 0 00 0 01CA 2C66 2 C66 = 14 @2E@ 2

T(12)=0B@

1 0 00 1 00 0 01CA

0B@

0 00 00 0 01CA C11 2 + C12 2 C12 = � @2E@ 2 � 2C11� =2

T(13)=0B@

1 0 00 0 00 0 11CA

0B@

0 00 0 00 0 1CA 12C11 2 + 12C33 2 + C13 2 C13 = � @2E@ 2 � C11 � C33� =2

Fig. 3. Comparison between the DFT calculation and themodel potential with (a) the potential parameters in Ref. 14and (b) the potential parameters in this study. is a scalarmeasure of the magnitude and sign of the strain � in Eq. (13).

Table 3. Elastic sti�ness constants for PbTiO3 in thisstudy compared with the literature values in units of GPa.This study Literature valuesDFT Ref. 23a Ref. 24a Ref. 25bC11 290 � 2 237 � 3 235 � 3 133C33 101 � 1 90 � 10 105 � 7 93C44 62 � 0 69 � 1 65 � 1 80C66 103 � 0 104 � 1 104 � 1 93C12 117 � 0 90 � 5 101 � 5 85C13 92 � 0 100 � 10 99 � 8 89aSingle-crystal Brillouin measurements.bLattice dynamics calculations using a rigid ion model.T(33) strain tensors are very large (Figure 3(a)). Thus,the system with the original parameter set will be un-stable at those strains. However, the new parameterset in Table 1 gives good agreement between the en-ergy di�erence from the DFT calculations and the en-ergy di�erence from the bond-valence potential model(Figure 3(b)). The previously reported potential param-eters are good for constant temperature molecular dy-namics simulations at room temperature. However, thephase transition temperature of the potential parameterset is smaller than the experimental one by about 200K.Also this potential parameter set is not good for con-stant stress simulations because the supercell breaks at�nite temperatures. These problems can be solved byadding more reference structures with a variety of lat-tice constants. The tetragonal-to-cubic phase transitiontemperature from the new parameter set in Table 1 is

-1210- Journal of the Korean Physical Society, Vol. 52, No. 4, April 2008�700 K, which is much closer to the experimental valuethan the value obtained by using the old parameters inRef. 14. Also, the new parameter set enables us to per-form constant pressure simulations because it considersenergies with di�erent cell shapes.

IV. CONCLUSIONSWe introduced an atomic potential model based on aphysical property (the inverse relation between the bondlength and the bond valence) for PbTiO3. To optimizethe parameters of this potential model, we used the sim-ulated annealing global optimization method. A scoringfunction was generated in terms of the energy di�erencesand the forces of reference structures, which were ob-tained from the DFT calculations. From the lattice con-stants and the displacements of ions, we could determiner��0o . The other potential parameters were obtained byusing the simulated annealing method to minimize thescoring function. By adding the strained reference struc-tures, we were able to reproduce the elastic properties ofPbTiO3 from DFT calculations in the atomic potentialmodel.

ACKNOWLEDGMENTSThis work was supported by the Brain Korea 21Project and by the U.S. Department of Energy GrantNo. DE-FG02-07ER46431.

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