Journal of Alloys and Compounds 574 (2013) 520–525

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Journal of Alloys and Compounds

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Structural, electronic, elastic and vibrational properties of BiAlO3:A first principles study

0925-8388/$ - see front matter Crown Copyright � 2013 Published by Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jallcom.2013.05.158

⇑ Corresponding author. Tel.: +90 322 3386661x2480; fax: +90 322 3386070.E-mail address: scabuk@cu.edu.tr (S. Cabuk).

Ulas Koroglu a, Suleyman Cabuk a,⇑, Engin Deligoz b

a Cukurova University, Faculty of Science and Letters, Physics Department, Adana 01330, Turkeyb Aksaray University, Faculty of Science and Letters, Physics Department, Aksaray 68100, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 February 2013Received in revised form 24 May 2013Accepted 24 May 2013Available online 1 June 2013

Keywords:BiAlO3

Density functional theoryElastic constantsPhonon dispersion curveElectronic structure

We present a study of the structural, electronic, elastic and vibrational properties of the rhombohedralBiAlO3 structure within the local density approximation of density functional theory using norm-con-serving pseudopotentials. The calculated equilibrium lattice constant, angle and atomic position are inreasonable agreement with the available experimental and theoretical dates. Based on the elastic con-stants and their related parameters, the crystal mechanical stability have been discussed. The elastic con-stants for BiAlO3 are also needed to completely determine its elastic properties including polycrystallinebulk, shear and Young’s moduli, Poisson’s ratio and the elastic anisotropy. Energy band structure showsthat the rhombohedral BiAlO3 has an indirect band gap between D and U-D symmetry points. We com-pute Born effective charge tensor, which is found to be quite anisotropic of Bi and O atoms. BiAlO3 havebeen studied by applying the direct method and deriving the phonon dispersion relations which includethe longitudinal/transverse optical phonon mode splitting. In the rhombohedral phase the phonon dis-persion curves show a soft mode between X and U-point. This soft mode leads to the observed rhombo-hedral-cubic phase transition. The results are compared with the previous calculations and availableexperimental data.

Crown Copyright � 2013 Published by Elsevier B.V. All rights reserved.

1. Introduction

Bismuth aluminate (BiAlO3) belongs to the perovskite familywith the general formula ABO3 and has the hexagonal (or rhombo-hedral) structure at room temperature with space group R3c (no:161) [1]. Perovskite oxides are important materials for variousfunctional devices. Perovskite-type oxides exhibit unique proper-ties such as ferroelectricity, piezoelectricity, and semiconductivity.Recently, Bi-containing perovskites and related perovskite-typecompounds have received tremendous attention as lead-free ferro-electric [2,3] and multiferroics [4,5] materials. BiXO3 (X = Al, Ga, Inand Sc) with nonmagnetic elements have traditionally been usedfor the modification of Pb-based piezoelectric materials. Beliket al. [1] prepared BiAlO3 using a high-pressure high-temperaturetechnique at 6 GPa and 1273–1473 K. It is reported that BiAlO3 isisotopic with multiferroic material like BiFeO3. Also they studiedthe vibrational properties of hexagonal BiAlO3 by Raman spectros-copy. In addition, BiXO3 compounds, where X is magnetic, i.e.,X = Fe, Mn, Co, Cr, etc., have recently received a lot of attentionas multiferroics [6–9]. Among simple BiXO3 compounds onlyBiAlO3 and BiInO3 have polar perovskite crystal structures. BiAlO3

exhibits many interesting properties with distinct structures atroom temperature. BiAlO3 crystallizes in a noncenterosymmetricR3c structure. Noncentrosymmetric crystals have important prop-erties such as ferroelectricity, phyroelectricity, piezoelectricity andsecond-order nonlinear optical properties. Wang et al. [10] investi-gated the structure, electronic properties, zone-center phononmodes, and structure instability of cubic BiXO3 compounds.Furthermore, the structural, electronic, elastic and optical proper-ties of cubic [10–13] and hexagonal [1,14,15] BiAlO3 have beeninvestigated using the first-principle (or ab initio) method. How-ever, many properties of the rhombohedral BiAlO3 are not studiedtheoretically.

In this study, the structural, elastic, electronic, Born effectivecharge and vibrational properties of the rhombohedral BiAlO3 aredetermined by using the first-principles density functional theory(DFT) with local density approximation (LDA). The elastic con-stants can provide valuable information about the anisotropiccharacter of the bonding and structural stability.

2. Computational details

In the trigonal crystal system, BiAlO3 can be structurally charac-terized by either a hexagonal or a rombohedral (primitive) unitcell. The rhombohedral unit cell contains two formula units for a

Fig. 1. Primitive unit cell of the rhombohedral phase of BiAlO3.

Table 1The optimized and experimental lattice parameters and WP positions of atoms in therhombohedral BiAlO3.

Reference a (Å) a (�) Atom, WPa Position

This work 5.3376 59.9898 Bi, 2a (0.0, 0.0, 0.0)Al, 2a (0.2321, 0.2321, 0.2321)O, 6b (0.5249, 0.4321, 0.9631)

Expt.a 5.4372 59.2502 Al, 2a (0.2222, 0.2222, 0.2222)O, 6b (0.4907, 0.4354, 0.9482)

Cal. [14] 5.4730 59.3800 Al, 2a (0.2741, 0.2741, 0.2741)O, 6b (0.5462, 0.0746, 0.9865)

a WP: Wyckoff position. Refs. [1,14].

U. Koroglu et al. / Journal of Alloys and Compounds 574 (2013) 520–525 521

total of ten atoms. The hexagonal coordinates, lattice constants andangles for BiAlO3 are converted to the rhombohedral coordinates.

The first-principles calculations are based on the DFT with theLDA. All calculations except for phonon dispersion curves are doneby using Spanish Initiative for Electronic Simulations with Thou-sands of Atoms (SIESTA) computer program [16,17]. The exchangecorrelation potential is described using the LDA [18]. The self-con-sistent ‘‘norm-conserving’’ pseudopotentials are generated usingthe Troullier–Martins scheme [19], in the Klienman–Bylander[20] fully nonlocal separable representation. The double zeta basisset was used in the calculations along with the 300 Ryd mesh cut-off for the grid. Brillouin zone(BZ) sampling of the crystal structurewas made by 12 k-points. Bi:6s2 6p3, Al:3s2 3p1 and O: 2s2 2p4 elec-tron configurations were considered as valence states in the con-struction of the pseudopotentials. During the calculations, wetook only the valence electrons of atoms. Because only these elec-trons are playing an important role in the physical properties of thecrystals. The cut-off radii for tested atomic pseudopotentials are ta-ken as s: 3.88 au, p: 2.77 au, d and f: 2.99 au for Bi, s: 1.86 au, p:2.06 au, d and f: 2.22 au for Al and 1.47 au for the s, p, d and f chan-nels for O. Atoms are allowed to fully relax according to the atomicposition and cell parameters with an accuracy fall below 0.005 eV/Å for forces. In self-consistent field calculation tolerance was calcu-lated to be 1 � 10�6 and described as the maximum difference be-tween input and output.

The phonon dispersion curve and density of states (DOS) ofBiAlO3 along the high-symmetry directions in the BZ are deter-mined using the direct method [21] and PHONON software [22].This software is compatible with SIESTA and the Hellmann–Feyn-man forces on atoms for generating the phonon dispersion andthe DOS. The Hellmann–Feynman forces are computed for six inde-pendent displacements, along x, y and z for Bi, Al and O atoms, asrequired by the rhombohedral supercell. We use a supercell2 � 2 � 2 times the conventional unit cell which consists of theprimitive unit cell. The phonon density of states was obtained byintegration of the phonon frequencies with a very high numberof k-points. The displacement amplitude is taken as 0.03 Å, to in-crease the positive and negative atomic displacements along x, yand z axes. The theoretical and application detail of PHONON Soft-ware can be found in PHONON manual and references therein [22].

3. Results and discussion

3.1. Structural properties

We determined the structural parameters BiAlO3 in the rhom-bohedral phase by relaxing simultaneously the cell shape andatomic position. The BiAlO3 structure in the rhombohedral unit cellis shown in Fig. 1. The calculations were done assuming convergedvalues for lattice constants, angle and atomic positions in thereduced coordinates in Table 1. During the structural optimiza-tions, we held the Bi atom fixed at the origin. The coordinates ofthe Al and O atom are reported in Table 1. The coordinates of theother atoms can be easily obtained by using symmetry operationsof the space group (R3c). All calculation of BiAlO3 have beenperformed using optimized lattice constants, angle and atomicpositions.

The calculated equilibrium lattice constant, angle and atomicpositions are listed in Table 1 to compare with the available exper-imental data [1] and other calculation [14]. It is clear that the cur-rent theory is in good agreement with experiment by evidence of amaximum deviation of 1.83% (lattice constant) and 1.25% (angle).Our results are very close to the experimental and other calcula-tions, which confirms the reliability of our approach.

3.2. Elastic properties

The elastic constants of the rhombohedral BiAlO3 give impor-tant information on its dynamical and mechanical properties. Itis well known that a rhombohedral crystal have six independentstress–strain coefficients (Bijkl(P)) [23]. Bijkl(P) are equal to Cijkl

when the pressure (P) vanishes. The mathematical form of theanisotropic properties of crystal is described using tensors.Although there are 81 terms in full elastic constant tensor, thenumber of independent terms is smaller. Use of the symmetry im-plicit in the Voight notation, namely that Cijkl = Cjikl = Cijlk = Cjilk re-duces the number of independent coefficients 36. We writesimply Cijkl = Cmn (i, j, k, l = 1, 2, 3; m, n = 1,. . ., 6) [24]. Within theVoight notation, we also have the symmetry that Cmn = Cnm, sothe fourth-rank tensor Cijkl generally comprises 21 independentcomponents. However, this number is greatly reduced when takinginto account the symmetry of the crystal. In a rhombohedral crys-tal, it is reduced to eight components, i.e. C11, C12, C13, C14, C33, C44,C65 and C66 [25]. Actually, only six independent elastic constantcomponents exist in the rhombohedral phase, since C65 = C14,C66 = 1/2(C11 � C12). The elastic constants are calculated using the‘‘volume-conserving’’ technique [26,27]. The elastic constants ofBiAlO3 are summarized together with the available hexagonal data

Table 2Elastic constants Cij (in GPa) and elastic bulk modulus (in GPa), Young moduli (in GPa), Shear modulus (in GPa) and Poisson’s ratio for the rhombohedral BiAlO3.

C11 C12 C13 C14 C33 C44 C65 C66 BV

This work 366 112 88 1.90 374 124 1.90 127 186.888Hex.a 301.66 118.85 34.85 300.86 68.54

BR B GV GR G E V B=GThis work 186.823 181.364 129.533 129.050 128.586 312.018 0.213 1.445Hex.a 140.55

a Ref. [15].

522 U. Koroglu et al. / Journal of Alloys and Compounds 574 (2013) 520–525

[15] in Table 2. To cross-check the calculation result C65 and C66

were also obtained and given in Table 2. The elastic constants ofC11 and C33 are significantly larger than the other elastic constants,resulting in a pronounced elastic anisotropy. All these elastic con-stants for the rhombohedral phase are positive and satisfy the gen-eralized Born criteria for mechanically stable crystals [25]:

ðC11 � C12Þ > 0; ðC11 þ C12Þ > 0; C33 > 0; C44 > 0

ðC11 þ C12ÞC33 � 2C213 > 0; ðC11 � C12ÞC44 � 2C2

14 > 0ð1Þ

Apart from the elastic constants, the bulk, shear and Young’s mod-ulus and Poisson’s ratio have been widely used for describing theelastic behavior of materials. These parameters for crystals can bedetermined using the elastic constants obtained in the first princi-ples calculations. In real applications, the mechanical properties ofsingle crystal of large size scales are not normally representative.Therefore, polycrystalline properties should be investigated to pro-vide beneficial valuable information in present applications ofBiAlO3. The polycrystalline material could be treated as an aggre-gate of single crystals at random orientation. The mechanical prop-erties of the polycrystalline material can be determined from thesingle crystal elastic constants using two main methods. These arethe Voight [28] and Reuss [29] methods, which lead to the asymp-totically maximum and minimum elastic parameters, respectively.In the Voight method [28] (BV and GV) and Reuss methods [29](BR and GR), the bulk and shear modules are written in the form

9BV ¼ 2ðC11 þ C12 þ 2C13Þ þ C33; ð2Þ

GV ¼1

30½C11 þ C12 þ 2ðC33 � 2C13Þ þ 12ðC44 þ C66Þ�; ð3Þ

BR ¼ðC11 þ C12ÞC33 � 2C2

13

C11 þ C12 þ 2C33 � 4C13; ð4Þ

GR ¼52

½ðC11 þ C12ÞC33 � 2C213�C44C66

3BV C44C66 þ ½ðC11 þ C12ÞC33 � 2C213�ðC44 þ C66Þ

ð5Þ

The Poisson’s ratio (V) and Young’s modulus (E) are frequently usedelastic parameters of polycrystalline materials when investigatingthe hardness of the solids. The arithmetic average of the Voigtand Reuss bounds is commonly used of estimate the polycrystallinemodulus. Using the Voigt–Reuss–Hill approximations [30], bulk andshear moduli can be written as:

B ¼ 12ðBV þ BRÞ and G ¼ 1

2ðGV þ GRÞ ð6Þ

The Young modulus and Poisson’s ratio were calculated usingthe B and G by the following equations:

E ¼ 9BG3Bþ G

and V ¼ 3B� 2G2ð3Bþ GÞ ð7Þ

The elastic parameters calculated using Eqs. (2)–(7) for BiAlO3 arelisted in Table 2. Bulk modulus measure the resistance that mate-rial offers to changes in its volume. From Table 2, we can see that

the bulk modulus of BiAlO3 is 181.364 GPa, comparable to the val-ues of 219 GPa [12] and 218.5 GPa [10] for the cubic and140.55 GPa [15] for the hexagonal BiAlO3 determined by the LDA.The bulk modulus of the rhombohedral structure is less than thatof the cubic structure, indicating that the cubic structure is lesscompressible than the rhombohedral structure. This is consistentwith the smaller volume of cubic structure than that of the rhom-bohedral structure. There is also a similar situation in BaTiO3

[172 GPa (cubic) and 129 GPa (rhombohedral)] [25]. The Young’smodulus, which measures the resistance of a material to changein its length, is the ratio of tensile stress and tensile strain and givesinformation about stiffness. The higher the Young’s modulus, thestiffer is the material. The shear modulus measures the resistanceto motion of the planes of a material sliding past each other. Thecalculated shear modulus of 128.586 GPa and Young’s modulus312.018 GPa in the rhombohedral BiAlO3 are less than the corre-sponding values of 139 GPa and 347 GPa in the cubic phase [12],respectively. B > G is clearly seen from Table 2; this implies thatthe parameter limiting the stability of BiAlO3 material is the shearmodulus (G). The calculated B=G is roughly considered as a criteriato judge the ductility properties, the critical value is 1.75, belowthat the material is regarded as brittle [31,32]. The calculated B=Gvalue for BiAlO3 is 1.445, suggesting that BiAlO3 is brittle. Perov-skite-type structures are usually classified as ionic materials. Assuch, lead titanate (PbTiO3) is sometimes considered in a simplemodel as a Pb2+ Ti4+ O2� crystal. The value of the Poisson’s ratio(V) is �0.1 for covalent materials, 0.25 for ionic materials and0.33 for metallic materials [33]. According to our calculation, Pois-son’s ratio is equal to 0.213 in the rhombohedral BiAlO3. Therefore,the ionic contribution to inter atomic bonding for this material isdominant. While Poisson’s ratio is 0.231 in the rhombohedralphase, smaller than 0.241 [12] in the cubic BiAlO3. The small Pois-son’s ratio indicates that the bonding is more directional. Most ofthe solids exhibit elastic anisotropies of varying degree. The anisot-ropy of solids can be obtained in several ways. The elastic anisot-ropy of materials reflects a different characteristic of bonding indifferent directions. The calculation of the elastic anisotropy isestablished in the crystal physics. For the rhombohedral structure,the shear anisotropy factors are given by [34].

A1 ¼2C66

ðC11 � C12Þfor the f100g orf010g shear plane ð8Þ

A2 ¼4C44

ðC11 þ C33 � 2C13Þfor the f001g shear plane ð9Þ

For an isotropic material, A1 and A2 must have value one, while anyvalue greater or smaller than one is a measure of the degree of elas-tic anisotropy possessed by the material. The calculated anisotropyfactors are listed in Table 3. As seen from Table 3, the rhombohedralBiAlO3 has small anisotropy on the {001} plane. Whereas the isot-ropy is very obvious on the {100} or {010} plane. Another way ofmeasuring the elastic anisotropy is called percentage elastic anisot-ropy [35]. The percentage of anisotropy in the compressibility (AC)and shear (AS) are defined as

Table 3The shear anisotropic factors A1, A2, AC (%) and AS (%).

A1 A2 ACð%Þ ASð%Þ

1 0.879 0.017 0.187

Fig. 2. The calculated band structure of the rhombohedral BiAlO3.

U. Koroglu et al. / Journal of Alloys and Compounds 574 (2013) 520–525 523

AC ¼ðBV � BRÞ

ðBV þ BRÞ� 100 and AS ¼

ðGV � GRÞ

ðGV þ GRÞ� 100 ð10Þ

For the percentage anisotropy, a value of zero indicates the elas-tic isotropy and a value of 100% is associated with the largestanisotropy [35]. The calculated values of the anisotropy percentageare also given in Table 3. It is seen in the R3c phase of BiAlO3 thatthe anisotropy in compression is very small and the anisotropy inshear is also small.

Fig. 3. Density of states and partial density of states of the rhombohedral BiAlO3.

3.3. Electronic properties

The calculated electronic band structure of the rhombohedralBiAlO3 in the high-symmetry directions in the BZ is shown inFig. 2. The Fermi level is set to zero and indicated by a horizontaldashed line. The lower bands which are located between�20.2 eV and �17.1 eV below the Fermi level (EF) are predomi-nantly of O 2s. The upper valance band with width of about6.0 eV is derived mainly from the O 2p states with some mixingwith Bi 6s states. It is clear that the indirect band gap appears be-tween the upper valence band at a point along C-D direction andthe bottom of the conduction band at the D point. The calculatedindirect band gap is 2.708 eV. This band gap value is between thecubic (1.76 eV) [12] and the hexagonal BiAlO3 (3.28 eV) [15]. Ourresult is smaller than the calculation value of the absorption edge(optical band gap) of 3 eV [14]. Note that there is no experimentaloptical band gap for comparison. The origin of this discrepancymay be the LDA which underestimates the energy band gaps evenfor insulators. The conduction bands have compounded primarilyfrom Bi 6p, O 2p and Al 6p states.

To further elucidate the nature of the electronic band structure,we have also calculated total density of states (DOS) and partialdensity of states (PDOS) of BiAlO3. These are displayed in Fig. 3.For BiAlO3, it is found that the upper valence bands (between�8.88 and 0 eV) are essentially dominated by O 2p states, with aminor admixture from Bi (6s, 6p) and Al (3s, 3p) states. There arehybridization between Al and O atoms and between Bi and Oatoms in this region, which suggests a covalent bonding contribu-tion in this material. The s, p states of Bi and Al atoms are also con-tributing to the valence bands, but the values of the correspondingdensities of states are quite small compared to O p states. The total

DOS of BiAlO3 shows a narrow band peaked around �10.9 eV; thisband is formed by the Bi s states with small contribution of O s, pand Al s states. Following Fig. 3, the lower band situated in therange �20.6 eV to �16.67 eV is due to an admixture from Al s, pand O s states. The bottom of the conduction band is dominatedby Bi p states which hybridize with O p states in the BiAlO3. Onthe other hand, there is an energy gap between the occupied O pstates and the unoccupied Bi p states, as seen from the analysisof the PDOS. Our band structure, DOS and PDOS calculation ofBiAlO3 agree well with other theoretical cubic [12] and hexagonalBiAlO3 [15].

3.4. Born effective charges

For insulators, the Born effective charge Z� play a fundamentalrole in the dynamics of the crystal lattice. The Born effective chargegovern the amplitude of the long-range Coulomb interaction be-tween nuclei and the splitting between longitudinal (LO) andtransverse (TO) optic phonon modes. The Born effective chargeare tensors, calculated for each atom by finite differences (SIESTAcode). For atom k, Z�k;ab created along the direction b when theatoms of sublattice k are displaced along the direction a. Theyare calculated as the change in the electronic polarization inducedby the small displacements generated for the force constants calcu-lation and defined as the first derivative of polarization (Pa) withatomic displacement (u)

Z�k;ab ¼V0

e@Pa

@uk;b

����q¼0

ð11Þ

Table 4Calculated Born effective charge (in atomic units) of Bi, Al, O1, O2 and O3 in therhombohedral phase BiAlO3.

Atom Nominal charge Effective charge tensor

Bi 3 5:1756 �0:2528 0:00:2528 5:1756 0:0

0:0 0:0 5:2259

0@

1A

Al 3 2:886 �0:1895 0:00:1895 2:886 0:0

0:0 0:0 2:9686

0@

1A

O1 �2 �2:8239 0:3086 �0:26110:3754 �2:5477 0:0116�0:2876 0:1769 �2:7960

0@

1A

O2 �2 �2:3218 �0:0820 0:12350:0148 �3:0018 �0:35840:0051 �0:2583 �2:8372

0@

1A

O3 �2 �2:8539 �0:1250 0:38850:3754 �2:6618 0:20490:2849 0:3559 �2:6343

0@

1A

Fig. 4. The calculated phonon dispersion curves along high-symmetry direction inBZ for BiAlO3.

Fig. 5. Phonon density of states of Bi, Al and O atoms and total density of state ofBiAlO3.

524 U. Koroglu et al. / Journal of Alloys and Compounds 574 (2013) 520–525

where V0 is the unit cell volume and e is the charge of an electron.As previously discussed in the literature [36], Z� is a dynamicalquantity, effecting orbital hybridization induced by the atomic dis-placements. In this regard, we calculated using SIESTA the Z� in therelaxed structure by finite differences of polarization for Bi, Al and Oatoms, which are given in Table 4. Due to the low symmetry, the fulltensors must be considered. The form of effective charge tensor forthe constituents is determined by the site symmetry of the Bi, Aland O atoms. The Bi cation, which has site symmetry (2a Wyckoffposition), is almost diagonal with anisotropy. While the nominal io-nic charge is between +5.1756 and +5.2259 for Bi. The Bi ion showsa strongly anomalous charge, more than almost double the nominalcharge. The Al has site symmetry (2a Wyckoff position). We ob-served that Z�Al is nearly isotropic and that the diagonal elementshave a value close to the nominal charge of the Al atom(+3), indicat-ing that it does not rehybridize through the ferroelectric phase tran-sition. The shape of Z� for O ion is markedly different from Bi and Alcations as can be seen from Table 4. O ions are located at lower sym-metry site (6b Wyckoff positions) and as a result their effectivecharge tensors have nonequivalent diagonal components as wellas sizable off – diagonal components. For the oxygen atoms, theanisotropy is much stronger. The result of Al atom seems to be quiteclose to the nominal charge but the Bi and O results show some dif-ferent anomalies from the nominal charge. Thus, the charge transferin BiAlO3 is seen that Al partly gain charges while Bi and O losecharge. This different behavior of BiAlO3 structure like the cubic BiI-nO3 [10] may be the cause of the structure difference from otherABO3 perovskites. There is only one theoretical study on the Borneffective charge of the cubic BiAlO3 (Z�Bi ¼ 6:22; Z�Al ¼ 2:84;Z�Ojj ¼ �2:34, Z�O? ¼ �3:38) [2] but this is not compatible one toone with our result in Table 4.

3.5. Phonon dispersion curves

Since the rhombohedral primitive unit cell of BiAlO3 structurecontains ten atoms, the complete phonon spectrum consists of30 dispersion curves: 3 acoustical branches and 27 opticalbranches. According to group theory, the rhombohedral structure(R3c) have the following phonon modes at the U point

C ¼ 5A1ðIR þ RÞ þ 5A210EðIR þ RÞ ð12Þ

where IR and R corresponds to infrared- and Raman- active modes,respectively. The A1 and E are Raman and infrared active so thatthey are split into transverse and longitudinal components. The A2

modes are not seen in Raman and infrared absorption, but are par-tially detected by inelastic neutron scattering. The TO phonon fre-quencies are calculated within the direct method. The LO modes

depend on the non-analytical term [37], which in turn dependson the effective charge and optical dielectric tensor. The calculatedBorn effective charge tensors and the optical dielectric constant [14]are used to determine the non-analytical term to phonon frequen-cies. For the rhombohedral structure, the phonon frequencies occurat the high-symmetry points X, U, A, Z and D. Fig. 4 shows the calcu-lated phonon dispersion curves along high-symmetry directionsand the corresponding phonon density of states (PDOS) are shownin Fig. 5. As seen in Fig. 4 the mode is a soft (imaginary) mode.The most interesting feature of phonon dispersion curves of rhom-bohedral BiAlO3 has been found to be unstable due to transverse

Table 5Comparison of the mode frequencies (THz) for BiAlO3 in the U point. The infrared-active modes A1 and E correspond to the transverse AT

1 and ET frequencies.

Mode This work Ref. [1] Mode This work Ref. [1]

E 2.581 1.38 E 12.015 12.33A2 3.348 E 12.615 15.36E 3.966 1.65 A2 13.347A1 6.063 3.60 E 14.028 17.22E 6.770 5.91 A2 14.46A1 7.845 7.92 E 15.955 18.51E 10.615 9.66 A1 19.488 19.53A2 10.817 E 20.446 20.97A1 11.392 11.73 A2 22.768

U. Koroglu et al. / Journal of Alloys and Compounds 574 (2013) 520–525 525

acoustic (TA) phonon mode near X point, indicating that TA mode isunstable under the ambient pressure. So the rhombohedral BiAlO3

is easily transforms to paraelectric structure under small externalpressure. The soft mode anomaly appears along X-U in the recipro-cal space which have no the experimental and theoretical data forcomparing with the present work, so our result is predictive. TAmode is stable in the Brillouin zone expect for along X-U direction.It is know that the soft mode frequency implies the anharmonic ef-fects and structural instability. This mode responsible for the ferro-electric phase (R3c) to paraelectric phase transition. Thediscontinuities seen at the U point are due to the LO/TO splitting[38]. The calculated phonon frequencies in the U point are listedin Table 5. Belik et al. [1] observed 13 Raman active modes(4A1 + 9E) whose wave numbers are expected without taking intoaccount transverse and longitudinal splitting of A1 and E modesdue to the non-centrosymmetric crystal structure. The convertedto units of these modes (1 cm�1 � 0.030 THz) are listed in Table 5,but unfortunately there are no experimental and theoretical datato compare our results one to one. From the PDOS, we know thatthe negative part corresponds to Bi vibrations only. The PDOS spec-tra show one peak for Bi vibrations in a region of low frequencies.The low-frequency motions are mainly due to Bi ions (<4 THz),while the phonon modes with frequencies over 4 THz are due toO and Al, with a major contribution of O atoms than Al atoms.

4. Conclusion

In conclusion, the structural parameters, elastic constants, elec-tronic structure, Born effective charge tensors and vibrationalproperties of BiAlO3 have been obtained by using the density func-tional theory with local density approximation. Our structureparameters (lattice constant, angle and atomic positions) are inagreement with the experimental data and previous calculations.The calculated elastic constants indicated that BiAlO3 is mechani-cally stable. The related polycrystalline elastic parameters likeshear modulus, Young’s modulus, Poisson’s ratio and anisotropicfactors are calculated. Through the value of the calculated B/Gand V, we concluded that the rhombohedral structure of BiAlO3

is brittle and ionic. The calculated electronic band structure showsthat the rhombohedral phase of BiAlO3 has an indirect band gap of2.708 eV. From the DOS and PDOS analysis, we conclude that Al–Oand Bi–O bond covalently. The Born effective charge tensors arequite anisotropic of Bi and O atoms. For some directions, the Borneffective charge is larger than the nominal charge indicating acovalent bonding between Al (Bi) and O bond. The results showthat BiAlO3 is an ionic material and strong covalent bonding exists

between Al 3p (and Bi 6p states) and O 2p states. We have revealedthe structure of the rhombohedral phase of frequencies of thezone-center model and the soft mode between X and U high-sym-metry points which is responsible for rhombohedral-cubic phasetransition. At present, there are not yet any calculations and exper-imental results in elastic constants, Born effective charge tensorsand phonon dispersion curves for the rhombohedral BiAlO3. There-fore, we hope that our calculations serve as a reference for futureexperiment study.

Acknowledgments

This work supported by Cukurova University under ProjectNumber FEF2011D13 and FEF2010BAP8. The authors are gratefulto SIESTA group that we used in our computations.

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