Elastic and piezoelectric properties of BaTiO3 at room temperature

Physica B 271 (1999) 343}347

Elastic and piezoelectric properties of BaTiO 3 at room temperature A. Khalal!, D. Khatib!,*, B. Jannot" !Laboratoire de Physique du Solide The& orique, Faculte& des Sciences, BP: 28/S, Universite& Ibn Zohr, 80000 Agadir, Morocco "Laboratoire de Physique, Equipe d'Optique des Mate& riaux, Faculte& des Sciences Mirande, BP 400, Universite& de Bourgogne, 21011 Dijon, France Received 6 April 1999

Abstract We have calculated the phonon dispersion curves of barium titanate (BaTiO ) at room temperature. A lattice 3 dynamical formalism using the shell model is used. This microscopic model includes the short-range interactions of axially symmetric type and the long-range Coulomb interactions and taking into account the electronic polarizability of constituent ions. Zone center phonon and a few phonons in the "rst Brillouin zone are used for "tting. The values of relevant parameters are critically analyzed. A calculation of the elastic and piezoelectric properties is presented and compared with the available experimental data. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Elastic properties; Piezoelectric properties; BaTiO ; Lattice dynamics; Shell model 3

1. Introduction Crystals of perovskite family, such as BaTiO , 3 PbTiO , KNbO , etc., have been of constant inter3 3 est in physics because some of these materials show ferroelectric behavior and undergo structuralphase transition [1,2]. BaTiO may be considered one of the most 3 studied crystals of this family. Above 1353C it is cubic and belongs to space group Pm3m (O1). At ) temperature below 1353C it is ferroelectric and its structure is P4mm (C1 ). If the temperature is 47 lowered further the crystals of BaTiO undergo 3 new structural transitions at 53C and 903C,

* Corresponding author. Tel.: 212-08-22-09-57; fax: 212-0822-01-00.

transforming to orthorhombic and rhomboedric symmetries, respectively. A number of researchers have studied the temperature-dependent vibration spectra of this materials [3,4] utilizing Raman and infrared spectroscopic techniques. There are, however, con#icting reports with regard to the interpretation of their experimental observation in relation to the applicability of the so-called `softmode theorya originally proposed by Cochran [5] and Anderson [6] independently in order to explain the anomalous dielectric behavior and the structural phase transition in the ferroelectric materials. The tetragonal form of BaTiO has the space 3 group P4 mm (C1 ) and the ions occupy the posi47 tion is indicated in Table 1. The room-temperature lattice parameters for the tetragonal cell are represented in Table 2.

0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 2 0 2 - 1

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A. Khalal et al. / Physica B 271 (1999) 343}347

Table 1 Index and positions of di!erent ions of BaTiO in the tetragonal 3 phase Ions

Index

Position

Ba Ti O z O y O x

1 2 3 4 5

(0, 0, 0) (0.5, u, 0.5) (0.5, 0.5, v) (0.5, 0, w) (0, 0.5, w)

Table 2 Values of the parameters of primitive cell of BaTiO at room 3 temperature Parameter

Measured value

Ref.

a c u v w

3.992 (As ) 4.036 (As ) 0.513 !0.023 0.487

[7] [7] [8] [8] [8]

Note: a and c are the lattice parameters and u, v and w are the relative displacements of Ti, O and O or O , respectively. z x y

In this work, we make a theoretical study of phonons of crystal BaTiO in the tetragonal phase. 3 The elastic and piezoelectric constants are also calculated. A shell model with the short-range interactions of axially symmetric type and the long-range Coulomb interactions and taking into account of the electronic polarizability of constituent ions, with some approximation are used to describe the above properties.

tronic polarizabilities of ions, long-range Coulomb forces and short-range axially symmetric forces. Following Born and Huang [9] the dynamical matrix is constructed in terms of atomic force constants / ( ll{ ) that are second derivatives of the ab kk{ crystal potential energy to displacement u (l ) of ion a k (l ) in a direction. The potential energy may be k written as the sum of long-range Coulomb interactions and short-range interactions. Usually, the short-range part is expressed as follows: U(r)"ke~jr,

(1)

c U(r)" , rn

(2)

where the constant k, j, c and n depend on the ion pairs in interaction and r is the inter-atomic distance. A more general approach, called the axially symmetric model, has been introduced to represent the short-range interactions and they involve the constants A and B , for each pair of ions (k, k@), kk{ kk{ de"ned as follows: e2 R2U(r) A " , (3) kk{ 2< Rr2 k{k 1 RU(r) e2 " , (4) B kk{ 2< r Rr kk{ kk{ where < is the primitive cell volume. The dynamical matrix is expressed as follows [10]: D"R#ZCZ!(T#ZCY) ](T#YCY)~1(TH5#YCZ),

2. Model description There are many phenomenological models to describe the lattice dynamics of ionic crystals, such as rigid-ion model [9], the shell model [10}13], the breathing shell model [14], etc. The use of the shell model is interesting because this model take into account of the electronic polarizabilities of the constituent ions of the crystal and it describes the phonon spectra. We have, therefore, restricted to the applicability of the shell model with the elec-

(5)

where R is the short-range interaction matrix, ZCZ, ZCY, YCY and >CZ are the long-range interaction matrices Z and Y are ionic and shell charges matrices, respectively, T is the interaction matrix between the k ion and the shell of the neighbor k@ ion and T is a matrix de"ned as follows: T (kk@/q)"S (kk@/q)#d d K , (6) ab ab ab kk{ k where k and k@ are ions in the primitive cell (k, k@" Ba, Ti, O , O , O ). a and b are the Cartesian 3 2 1 coordinates and q is a vector in the "rst Brillouin

A. Khalal et al. / Physica B 271 (1999) 343}347

zone. K is the core shell coupling for k ion and k S the interaction matrix between the shells of neighbor ions. The eigenvectors of the dynamical matrix are the normal modes, whereas the eigenvalues are the frequencies of the lattice vibrations. The number of the independent parameters may be reduced further by assuming that the shortrange radial force constants are approximated by a Born}Mayer potential, they are given by e2 A"kj2e~jr. 2<

(7)

The coe$cients j can be related to the exponential factor n appearing in Pauling's potential in the following manner:

345

the case of BaTiO there are "ve structural para3 meters, namely a, c, u, v and w, and therefore, "ve stability conditions may be written, thus reducing the number of unknown short-range-potential constants (B1s) from nine to four. The resulting equation is written as

K

RU "0, (17) Rx 0 where x can be one of the lattice parameters a and c or one of the ion positions u, v and w along the [0 0 1] direction for the ions Ti, O and O , respec3 1 tively. The derivative is evaluated at the equilibrium position of the crystal. This equation can be used to determine "ve parameters, namely B , B@ , B , B@ and B . 24 13 14 23 23

[n (n #1)]1@2 , j "j } " 1 1 1 B! O r 14

(8)

3. Elastic and piezoelectric properties

[n (n #1)]1@2 j "j } " 2 2 , 2 T* O r 23

(9)

A propagate equation for the acoustic wave in piezoelectric medium along the [1 0 0] direction is written as follows [15]:

[n (n #1)]1@2 . j "j } " 3 3 3 OO r 45

(10)

The use of the above equations result in the reduction of radial force constants A from nine to three kk{ as follows: A "A exp[!j (r !r )], 13 14 1 13 14

(11)

A@ "A exp[!j (r@ !r )], 14 14 1 14 14

(12)

A@ "A exp[!j (r@ !r )], 23 23 23 2 23

(13)

A "A exp[!j (r !r )], 24 23 2 24 23

(14)

A "A exp[!j (r !r )], 34 45 3 34 45

(15)

A@ "A exp[!j (r !r )]. 34 45 3 34 45

(16)

The number of parameters B can be reduced kk{ by considering the stability conditions for the unit cell. This requires the precise knowledge of structural parameters of the crystal. The potential energy of the primitive cell is minimized with respect to each structural parameter. In

R2u R2u RE i"c k!e p, (18) 1i1k p1i Rt2 Rx2 Rx 1 1 R2u R2E R RE k !e@ p"k q e@ (19) 0 Rt p1iRtRx qp Rt Rx2 1 1 where c@ and e@ are the elastic and piezoelectric p1i 1i1k constants, respectively, when applying the external electric "eld. o is the density of the crystal, k is the 0 magnetic permeability and E represented the pth p component of the applied electric "eld. Solutions of the plane-wave type are valid for Eqs. (18) and (19): o

A

B

(20) u "u0e+(ut`kx), i i (21) E "E0e+(ut~kx), i i where u0 and E0 are the amplitudes of the wave i i shift and applied electric "eld, respectively. This choice of solutions give the following linear equations: ou2u0"k2c@ u0!je@ kE0, p1i p 1i1k k i k2E0"jk ku2e@ u0!k u2e@ E0. 0 qp q p1i i p 0

(22) (23)

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In the absence of an applied electrical "eld, the solution is obtained by solving only Eq. (18). From Eqs. (18) and (22) we have: (24) ou2u0"k2c@ u0. 1i1k k i This equation corresponds to the relationship between the elastic and piezoelectric properties and the acoustical modes of vibration.

4. Results and discussions In Table 3, we report the calculated values of the ionic and the shell charges of the primitive cell of BaTiO at room temperature. 3 The analysis of Table 3 shows a good agreement between the calculated ionic charge and the shell charge of various atoms of BaTiO at room tem3 perature and the experimental results [17]: Z " B! 1.480, Z "2.720 and Z " !1.400. T* O In Table 4, we report the calculated values of the electronic polarizabilities and the coupling core-shell constants for various atoms of BaTiO at 3 room temperature. The analysis of Table 4 shows that both electronic polarizabilities and coupling core-shell conTable 3 Calculated values of ionic and shell charges for ions of BaTiO 3 at room temperature Ion

Ba Ti O

Ionic charge Z (e) Present work

Previous work [16]

1.546 2.732 !1.420

1.667 3.253 !1.640

stants of oxygen ions are highly anisotropic. This result can be interpreted by the increase of the oxygen polarizability due to the hybridization of the p orbital of the oxygen ion with the d orbital of the Ti ion [18]. The calculated values of Paulings constants are: n "5.99, n "3.48 and n "7.00. 1 2 3 In Table 5, we report the calculated values for various parameters of the short-range interactions. The analysis of Table 5 shows that the shortrange interaction between Ti ion and the nearest oxygen is stronger than other interactions. By using the various values of the parameters reported in Tables 3}5, we can determine the dispersion curves of the vibration normal modes of BaTiO at room temperature. 3 Table 5 Values of the short-range force constants of BaTiO at room 3 temperature A r 13 r 14 r@ 14 r 23 r@ 23 r 24 r 34 r@ 34 r 45

kk{

(e2/2V)

4.616 5.200 4.377 183.000 99.166 129.356 !1.643 !1.514 !1.700

B (e2/2V) kk{ 8.991 !3.370 !3.109 !36.118 !13.452 !49.028 0.890 !0.700 0.950

Shell charge > (e)

!4.00 !0.44 !2.62

Table 4 The calculated values of electronic polarizabilities and coupling core-shell constants of BaTiO 3 Ion

a (As ~3)

K (e2/V)

Ba Ti O z

1.232 0.016 a "2.415 OB a "1.745 OA

1620.0 1400.0 K "140.0 OB K "450.0 OA

Fig. 1. Dispersion curves of transversal modes along the [1 0 0] and [0 0 1] directions of BaTiO at room temperature (v: 3 calculated frequencies and m: measured frequencies [19]).

A. Khalal et al. / Physica B 271 (1999) 343}347

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was taking into account the anisotropic of the electronic polarizability of the oxygen ions. A good agreement between the calculated and measured phonon frequencies is obtained. The resulting parameters showed that the short-range interaction between Ti ion and nearest oxygen is stronger than other interactions. The calculated elastic and piezoelectric constants are in fairly good agreement with the experimental data.

References Fig. 2. Dispersion curves of longitudinal modes along the [1 1 0] and [1 0 0] directions of BaTiO at room temperature 3 (v: calculated frequencies and m: measured frequencies [19]). Table 6 Elastic (1011 N/m2) and piezoelectric (C/m2) constants of BaTiO at room temperature 3

c 11 c 33 c 12 c 13 c 44 c 66 e 15 e 31 e 33

Calculated values

Measured values [20]

1.67 1.26 1.25 1.21 0.52 0.93 20.37 !3.57 8.45

2.22 1.51 1.34 1.11 0.61 1.34 34.2 !0.7 6.7

The calculated results along the [1 0 0], [0 0 1] and [1 1 0] direction in the "rst Brillouin zone are reported in Figs. 1 and 2. Both Figs. 1 and 2 show a good agreement between the calculated values and the experimental ones. In Table 6 we report the calculated values of the elastic and the piezoelectric constants of BaTiO at 3 room temperature.

5. Conclusion The shell model was applied to the ferroelectric crystal BaTiO in the tetragonal phase. This model 3

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