[Article] www.whxb.pku.edu.cn

物理化学学报(Wuli Huaxue Xuebao)

Acta Phys. -Chim. Sin. 2015, 31 (6), 1059-1068June

Received: February 14, 2015; Revised: April 12, 2015; Published on Web: April 14, 2015.∗Corresponding authors. TIAN Guang-Shan, Email: tiangs@pku.edu.cn; Tel: +86-10-62754231. JING Xi-Ping, Email: xpjing@pku.edu.cn;

Tel: +86-10-62754188.

The project was supported by the National Natural Science Foundation of China (21071009, 21371015).

国家自然科学基金(21071009, 21371015)资助项目

© Editorial office of Acta Physico-Chimica Sinica

doi: 10.3866/PKU.WHXB201504144

基于第一性原理计算的铁电材料BaTiO3相关振动光谱指认

安 炜 1 刘天慧 2,3 王春海 2 刁传玲 2 罗能能 2 刘 勇 2,3

戚泽明 4 邵 涛 4 王玉银 4 焦 桓 3 田光善 1,* 荆西平 2,*

(1北京大学物理学院, 北京 100871; 2北京大学化学与分子工程学院, 稀土材料化学及应用国家重点实验室,

北京 100871; 3陕西师范大学化学化工学院, 陕西省大分子科学重点实验室, 西安 710119;4中国科技大学, 国家同步辐射实验室, 合肥 230029)

摘要: 通过固相反应烧结法在1400 °C下烧结4 h合成了BaTiO3陶瓷, 并用X衍射确定了其为四方晶系. 进行

了拉曼谱和红外谱的测量, 并采用洛仑兹函数以及四参数方法分别对上述光谱进行了拟合. 基于第一性原理的

计算, 并考虑了横模纵模劈裂, 对拉曼和红外光谱进行了指认. 为了更好地分析振动模式, 所有振动模用群论的

对称坐标进行了分解. 在12个光学模中, 仅具有拉曼活性的B1模式是O4和O5沿着z轴的相对运动. A1模式和

E1软模式是从立方BaTiO3相的F1u模式劈裂出来的, 对于四方相BaTiO3的铁电性有着重要作用, 其体现在A1(1)

模式造成了铁电z轴的极化, E1模式导致了大介电常数. 这两个模式都可以看成是Ti原子相对于O6八面体笼子

沿着z轴或者是xy平面的振动.

关键词: 第一性原理计算; 拉曼模式; 红外模式; 介电性质

中图分类号: O649

Assignment for Vibrational Spectra of BaTiO3 FerroelectricCeramic Based on the First-Principles Calculation

AN Wei1 LIU Tian-Hui2,3 WANG Chun-Hai2 DIAO Chuan-Ling2 LUO Neng-Neng2

LIU Yong2,3 QI Ze-Ming4 SHAO Tao4 WANG Yu-Yin4

JIAO Huan3 TIAN Guang-Shan1,* JING Xi-Ping2,*

(1School of Physics, Peking University, Beijing 100871, P. R. China; 2State Key Laboratory of Rare Earth Materials Chemistry

and Applications, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, P. R. China;3Key Laboratory of Macromolecular Science of Shaanxi Province, College of Chemistry and Chemical Engineering,

Shaanxi Normal University, Xi'an 710119, P. R. China; 4National Synchrotron Radiation Laboratory,

University of Science and Technology of China, Hefei 230029, P. R. China)

Abstract: A BaTiO3 ceramic was synthesized using a conventional solid-state reaction, and sintered at 1400 °C

for 4 h. The pure tetragonal phase was confirmed by Rietveld refinement of the X-ray diffraction data. The

Raman spectrum and the far infrared (FIR) reflective spectrum were obtained and analyzed using Lorentz fitting

and the four-parameter semi-quantum model fitting, respectively. The Raman and FIR spectra were assigned

based on first-principles calculations, and consideration of the splitting of the transverse optical modes and

longitudinal optical modes. All the vibrational modes were represented by linear combinations of the symmetry

coordinates deduced by group theory analysis. Among the 12 optical modes, the Raman-active-only mode, B1,

can be viewed as the wing-flapping vibration of the O4-O5 plane perpendicular to the z-axis in the O6

octahedron. The A1(1) mode and the E(1) soft mode are split by the triply degenerate F1u mode of cubic BaTiO3,

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Acta Phys. -Chim. Sin. 2015 Vol.31

1 IntroductionInvestigation on solid state dielectric materials is one of the

most important research fields in material sciences.1 Due to the

development of the microwave communication technology, the

microwave dielectric ceramics, which can be used in resonator,

filter, antenna and so on, have drawn many attentions in the past

two decades.2 Based on the dielectric theory, the vibrational modes

of the dielectric ceramics have significant influence on their di-

electric properties, thus deep understanding on the vibrational

modes of the materials is very important. To study the influences

of the vibrational modes on the dielectric properties, far infrared

(FIR) reflective spectra were firstly analyzed using the four-

parameter semiquantum (FPSQ) model by Petzelt3 and Wakino4

et al. in 1980s, referring to the materials K2SeO4, Ba(Zn,Ta)O3-

BaZrO3 and so on, and these authors indicated the mode contri-

butions to the dielectric properties for these materials. Subse-

quently using three-parameter or the FPSQ models,5 Fukuda et al.6

have fitted the FIR vibrational spectra of BaSm2Ti5O14, BaTi4O9,

Ba(Zn1/3Nb2/3)O3 and so on. Henceforth, based on the same

approach, many microwave dielectric ceramics, such as

Ba6-3xSm8+2xTi18O54,7 CaSmAlO4,8 (Mg1-xZnx)Al2O49 and so on have

been investigated.10,11 However, in all the above work, authors did

not give the descriptions of the vibrational modes and did not es-

tablish the links between these vibrational patterns and dielectric

properties. In our previous work, we have studied the Raman and

FIR spectra for some dielectric materials, e.g. Ba(Mg1/3Ta2/3)O3,12,13

MgTiO3,14,15 Ba(Mg1/2W1/2)O3,16 and Ba(Mg1/3Nb2/3)O3.17 We not only

analyzed the mode contributions to the dielectric properties for

these materials, but also assigned and described the vibrational

patterns based on the first-principles calculations.

Ferroelectrics are one type of the special dielectric materials,

having various applications in electronics.18,19 BaTiO3, the proto-

type of ferroelectrics, has already attracted lots of researchers to

discover the origin of its ferroelectric properties and the mecha-

nism behind the para-ferroelectric transformation.20-22 Among

these reports, investigating correlations between the vibrational

modes and the ferroelectric properties is one of the important

aspects. By using FIR reflective spectrum, Luspin et al.23 have

assigned the soft mode in cubic phase (high temperature poly-

morph) and further verified that the phase transition down to

tetrahedron phase is essentially displacive type, rather than the

order-disorder type. The Raman spectrum of the single crystal

BaTiO3 (tetragonal phase) has long been measured and assigned

based on the symmetry analysis by DiDomenico et al.24 Besides,

in Wada's recent work,25 the FIR spectra of the polycrystalline

BaTiO3 materials (both powder and colloidal crystals) have been

measured and fitted by the FPSQ model to extrapolate the di-

electric behavior in terahertz (THz) region. Also, with the advance

of the density functional theory (DFT) first-principles calculation

approach, lots of calculating work for BaTiO3 has been conducted

on some aspects, such as surface structure relaxation, defect and

impurity, nano BaTiO3 structure and so on.26-30 In Evarestov's very

recent work,31 the frequencies of vibrational modes were calcu-

lated by various of functionals in DFT framework and the as-

signments to the Raman and FIR spectra were conducted ac-

cording to their symmetries. However, the studies combining the

vibrational mode calculations with the assignments of the mea-

sured Raman and FIR spectra are rare to the best of our knowl-

edge. Although studies both on the measurements of the Raman

as well as FIR spectra and the theoretical calculations for the

vibrational modes can be found, but normally the measuring work

and calculating work were conducted separately by different

authors. Additionally, the researches on describing the vibrational

patterns of BaTiO3 and establishing the relations between the

vibrational patterns and the ferroelectric behavior are still scarce.

In the present work, the methods we previously used12-17 for the

microwave dielectric ceramics, were applied for the ferroelectric

ceramic BaTiO3. Here we synthesized the BaTiO3 ceramic sample

with the pure tetragonal phase. The symmetry coordinates of the

material were obtained through group theory analyses and the

vibrational modes were calculated by the first- principles calcu-

lations. And then the vibrational modes were represented by the

linear combinations of the symmetry coordinates and the vivid

cartoon descriptions of all the vibrational modes were illustrated.

The Raman and FIR spectra were analyzed by Lorentz fitting and

the FPSQ model, respectively. Based on the calculated frequencies

of the transverse optical modes, the vibrational peaks in the

Raman and FIR spectra were assigned. The contributions of the

assigned modes to the dielectric properties were calculated and

analyzed combining the real space vibrational patterns. The results

show that the E(1) soft mode (24.0 cm-1) is mainly responsible to

the large permittivity of BaTiO3 and the A1(1) mode (177.6 cm-1)

causes its ferroelectric polarization.

2 Experimental and theoretical methodThe BaTiO3 ceramic sample was synthesized by conventional

solid state reactions. BaCO3 (99%, analytical reagent, AR, Beijing

Fine Chemical Company, China) and TiO2 (99%, analytical re-

agent, AR, Beijing Chemical Reagent Company, China) were used

as starting reagents, which were weighed in appropriate ratio and

mixed by grinding with an agate mortar and pestle. The ground

powder was firstly calcined at 1200 °C for 2 h, and then sintered

on a Pt plate at 1400 °C for 4 h. Prior to the sintering, the powder

was re-ground and pressed into a disk. The sintered disk has the

resulting in the ferroelectric property of tetragonal BaTiO3. The appearance of the A1(1) mode leads to crystal

polarization along the z-axis and the E(1) mode causes the large permittivity. These two modes can be described

as vibration of the Ti atom against the O6 octahedral cage along the z-axis [A1(1)] and on the xy-plane [E(1)].

Key Words: First-principles calculation; Raman mode; IR mode; Dielectric property

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AN Wei et al.: Assignment for Vibrational Spectra of BaTiO3 Ferroelectric CeramicNo.6

relative density of 98% measured using the Archimedes principle.

A Rigaku D/max- 2000 X- ray diffractometer (XRD, Tokyo,

Japan) with Cu Kα source, operated at 100 kV and 40 mA, was

used for the phase identification. The pattern was recorded in the

2θ range of 5°- 90° with a step size of 0.02° and a scan rate of 6 s∙step-1. The cell parameters and atom coordinates were refined by

Rietveld refinement using the software TOPAS (version 3).32 A

Horiba/Jobin Yvon laser Raman analyzer LabRAM HR 800

(Longjumeau, France) was used for Raman measurement (100-

1000 cm-1), in which a He-Ne laser source with the wavelength of

632.8 nm and a grating with 1800 line∙mm-1 was applied. FIR

reflective spectrum in the range from 100 to 4000 cm-1 was col-

lected on a Bruker IFS 66 (Karlsruhe, Germany) Fourier trans-

form infrared spectrometer, in which a Globar source, a 2 cm-1

resolution, and a 12° incident angle were applied. The spectrum

was recorded in vacuum and an Au mirror was used as a reference

during the data collection process. For the FIR spectrum mea-

surement, one surface of the disk was carefully polished with fine α-

Al2O3 powders until mirror-like surface was achieved. Dielectric

properties in microwave frequency were determined using the

method suggested by Hakki and Coleman33 and modified by

Courtney34 using a network analyzer (N5230A, Agilent, Palo Alto,

CA). Alternating current (AC) impedance analyzer (6500B,

Wayne Kerr Inc., Shenzhen, China) was employed for measuring

the dielectric properties in the frequency range of 1 Hz-1 MHz.

Prior to the measurements, the Pt electrodes on the two opposite

surfaces of the disk were prepared by coating the Pt paste with

two Pt wires following 800 °C heating for 10 min to burn off the

organic components in the paste. All the measurements were

carried out at room temperature.

The first-principles calculations within the framework of the

density functional theory were performed using the CASTEP

packaged35 implemented in Materials Studio 5.0. The local density

approximation (LDA) was used throughout all the calculations.

The norm-conserving pseudopotential in the CASTEP database and

the valence electron configurations of 5s25p66s2, 3d24s2, and

1s22s22p4 for Ba, Ti, and O were selected, respectively. To ensure

the convergence of the ground-state total energy, the cutoff energy

of the plane-wave basis was set to be 650 eV and the 6×6×6

Monkhurst-Pack k-mesh with (0.25, 0.25, 0.25) origin shift was

adopted. We set the convergence tolerance as 10-8 eV∙atom-1. For

geometrical optimization, both the cell parameters and internal

atomic coordinates were relaxed, using a force tolerance of 0.01

eV∙nm-1. The frequencies and the atom movements of the optical

vibrational modes can be obtained from the phonon calculation

based on the linear response method using the tetragonal unit cellof BaTiO3.

3 Results and discussionThe XRD pattern of the BaTiO3 ceramic sample and the Riet-

veld refinement results are illustrated in Fig.1. The TOPAS re-

finement was carried out based on the tetragonal structure model

of BaTiO3 (ICSD # 161340)36 with a space group P4mm (No. 99)

and the cell parameters a=0.39988(2) nm and c=0.40222(4) nm.

In this database model, all atom (Ba 1a, Ti 1b, O1 1b and O2 2c)

are at the special sites. Except the coordinates of the Ba atom (1a

site), the coordinates of all other atoms were refined and they are

given as: Ba 1a (0.0000, 0.0000, 0.0000), Ti 1b (0.5000, 0.5000,

0.4973), O1 1b (0.5000, 0.5000, 0.9604), and O2 2c (0.5000,

0.0000, 0.4928). Other refined results are listed in Table 1. The

satisfied refinement quality confirmed that the BaTiO3 ceramic

sample is a pure phase.

The refined cell parameters and the atom coordinates men-

tioned above were used as the initial structure for the geometric

optimization. The optimized cell parameters are a=0.40089074

nm, c=0.40089073 nm and the optimized atomic coordinates are

Ba 1a (0.000000, 0.000000, 0.000000), Ti 1b (0.500000,

0.500000, 0.502872), O1 1b (0.500000, 0.500000, 0.00276725),

and O2 2c (0.500000, 0.000000, 0.50275432). Compared with the

ICSD database and the previous DFT calculation in literature,31

our results are verified and indicate that the LDA functional is

able to describe the crystal structure. We have also tested the

Fig.1 Observed (cycles) and calculated (line) XRD patterns, as well as difference profile (line below the pattern), for

the Rietveld structure analysis of the BaTiO3 sample (color online)

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Acta Phys. -Chim. Sin. 2015 Vol.31

generalized gradient approximation (GGA) functionals which

show slightly large lattice parameter with a relative difference of

1.2%, therefore, to simplify the calculation, the LDA functional

was used afterwards. The tetragonal BaTiO3 structure is depicted

in Fig.2, in which eight Ba atoms are located at the corners of the

cell and a TiO6 octahedron is located at the center. There are five

types of vibrational atoms labeled from 1 to 5 constituting the

BaTiO3 formula. Except the O2 2c site, each structural atom site

is equivalent to one type of the vibrational atoms: Ba 1a➝1, Ti

1b➝2, O1 1b➝3; while the O2 2c site is split into two types of

the vibrational atoms: 4 and 5. The vibrational mode parameters

were calculated with the above geometrically optimized structure

parameters.

BaTiO3 unit cell contains 5 atoms, leading to 15 vibrational

modes. Excluding 3 acoustic modes, there are 12 optical modes.

According to the group theory analysis implemented in the

Billbao Crystallographic Center Website (www.cryst.ehu.es),37 the

12 optical vibrational modes of BaTiO3 can be represented as

follows:

Γopt=3A1(R, IR)+B1(R)+4E(R, IR) (1)

where R and IR denote Raman and IR modes, respectively, while

A1 and B1 denote the non-degeneracy modes, E denotes the double

degeneracy modes. If we consider double degeneracy mode E as

one mode, the BaTiO3 optical vibrational modes include 8 Raman

active modes (3A1, 1B1, 4E) and 7 IR active modes (3A1, 4E). In

addition, another A1 mode and a double degenerate E mode con-

stitute the acoustic modes of BaTiO3. In order to make the dis-

cussion concisely, we adopt one index number to distinguish the

different modes in the same symmetric category [e.g. 4E(R, IR)

modes were represented as E(1), E(2), E(3), and E(4)], and two index

numbers to distinguish each submode of the degeneracy modes

[e.g. E(1)=E(11)+E(12)]. For simplicity, we would denote“the optical

vibrational mode”as“vibrational mode”or“mode”in the fol-

lowing context. Also, since we only care about the normal mode

(q=0), the“vibrational mode”or“mode”means normal mode.

To further scrutinize the above vibrational modes, we employ

the symmetry coordinates for each in-equivalent type of the vi-

brational atoms. The Cartesian coordinates of the displacements

for each atom are denoted as xi, yi, and zi, where the subscript i is

consistent with the atom number shown in Fig.2. The symmetry

coordinates are represented by the lower case letters. The different

non-degenerate symmetry coordinates are distinguished by one

index number, e.g. a1(j), while the doubly degenerate symmetry

coordinates are denoted by two index numbers, e.g. e(j) = e(j1) + e(j2).

The (unitary) normalized transformations between the Cartesian

coordinates and the symmetry coordinates are represented by

equations from (2.1) to (2.15) and their vividly schematic illus-

trations are shown in Fig.3.

a(1)

1 (Ba) = z1 (2.1)

a(2)

1 (Ti) = z2 (2.2)

a(3)

1 (O) = z3 (2.3)

a(4)

1 (O) = 12

(z4 + z5) (2.4)

b1(O) = 12

(-z4 + z5) (2.5)

e(11)

(Ba) = x1 (2.6)

e(12)

(Ba) = y1 (2.7)

e(21)

(Ti) = x2 (2.8)

e(22)

(Ti) = y2 (2.9)

e(31)

(O) = x3 (2.10)

e(32)

(O) = y3 (2.11)

e(41)

(O) = x4 (2.12)

e(42)

(O) = y5 (2.13)

e(51)

(O) = x5 (2.14)

e(52)

(O) = y4 (2.15)

With the aid of the symmetry coordinates, the vibrational

modes for both Raman and IR active can be further represented

by the linear combination with differently weighted symmetry

coordinates within the same symmetric category,12-17 shown in the

transformation matrices from (3) to (5) ((5) in Scheme 1). The

Table 1 Crystallographic data of BaTiO3 derived from

the Rietveld refinement of XRD data

Parameter

formula

crystal system

space group

a/nm

c/nm

cell volume/nm3

formula units per cell Z

structure refinement

radiation

T/K

2θ range/(°)

Number of data points

profile function

crystal density/(g∙cm-3)

R-factor: Rp

Rwp

Rexp

GOF

χ2

R-Bragg

Value

BaTiO3

tetragonal

P4mm (No. 99)

0.39932(1)

0.40295(7)

0.0642546(3)

1

Rietveld refinement/Topas

Cu Kα

293

5-90

4251

PV_TCHZ

4.035

7.69

10.70

5.40

1.98

3.92

1.918

Fig.2 Tetragonal structure of BaTiO3 (color online)

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AN Wei et al.: Assignment for Vibrational Spectra of BaTiO3 Ferroelectric CeramicNo.6

A1 and E modes involve the displacements of all the atoms in

BaTiO3, while the B1 mode refers to the displacements of the O2

atom only. Combining all the symmetry coordinates with heavier

weights and ignoring the lighter ones for each vibrational mode,

we can schematically illustrate the main feature of each vibra-

tional mode. All the 12 modes are represented in Fig.4, in which

the direction and length of the arrow stand for the maximum

displacement of vector for each atom. It is the same as the sym-

metry coordinates a1 and b1 that all the non-degenerate modes A1

and B1 are polarized along z direction, while all the degenerate E

modes vibrate on xy-plane consistent with their e symmetry co-

ordinate components. In general, the displacements of the Ba and

Ti atoms are small due to their larger mass and the movements of

the O atoms are responsible for the most of the vibrations.

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A(1)

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A(2)

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A(3)

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ø

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-0.0145 -0.0777 0.1125 0.1731-0.0527 0.1027 0.0441 0.0714-0.0016 -0.0003 0.2088 -0.1374

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(1)

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a(2)

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a(3)

1 (O)

a(4)

1 (O)

(3)

B1 = -0.2500∙b1(O) (4)

It is worth mentioning that the Raman and IR active modes are

mutually exclusive for the materials with centrosymmetric space

groups, while for the materials with non-centrosymmetric space

groups, some of the Raman and IR active modes are mutually

inclusive.38 BaTiO3 has non-centrosymmetric space group (P4mm),

thus its A1 and E vibrational modes are active for both Raman and

IR, which is quite different from the materials we previously

studied, such as Ba(Mg1/3M2/3)O3 (M=Ta, Nb) (P3m1),12,13,17 MgTiO3

(R3),14,15 and Ba(Mg1/2W1/2)O3 (Fm3m)16 in which their Raman and

IR modes are exclusive due to their centrosymmetric space group.

In other words, the Raman active modes would be polar vibrations

or non-polar vibrations, but the IR modes must be polar ones.

Fig.4 clearly shows that both the A1 and E modes of the tetragonal

BaTiO3 are polar vibrations, but the B1 mode is Raman active only,

due to its non-polar feature.

Based on the above discussion, we know that all the modes of

BaTiO3 are Raman active ones. We measured Raman spectrum of

the BaTiO3 ceramic sample in the range from 70 to 1000 cm-1,

Fig.3 Sketches of the symmetry coordinates (color online)

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÷E (11)

E (12)

E (21)

E (22)

E (31)

E (32)

E (41)

E (42)

=

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÷-0.0046 -0.0135 -0.0254 -0.0740 0.0397 0.1154 0.0397 0.1155 0.0365 0.10630.0135 -0.0046 0.0740 -0.0254 -0.1154 0.0397 -0.1155 0.0397 -0.1063 0.0365

-0.0030 -0.0527 0.0058 0.1021 0.0029 0.0510 0.0029 0.0511 0.0025 0.04460.0527 -0.0030 -0.1021 -0.0058 -0.0510 0.0029 -0.0511 0.0029 -0.0446 0.00250.0000 0.0000 0.0000 -0.0000 0.1763 -0.0145 -0.1761 0.0145 0.0001 0.00000.0000 0.0000 0.0000 -0.0000 -0.0145 -0.1763 0.0145 0.1761 0.0000 -0.00010.0016 0.0000 0.0003 -0.0000 0.0972 0.0000 0.0972 0.0000 -0.2088 0.00000.0000 0.0016 0.0000 -0.0003 0.0000 0.0972 0.0000 0.0972 0.0000 -0.2088

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÷e(11)(Ba)

e(12)(Ba)

e(21)(Ti)

e(22)(Ti)

e(31)(O)

e(32)(O)

e(41)(O)

e(42)(O)

e(51)(O)

e(52)(O)

(5)

Scheme 1 Transformation matrice (5)

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Acta Phys. -Chim. Sin. 2015 Vol.31

illustrated in Fig.5, which is in a good agreement with other lit-

erature data for polycrystalline, microcrystalline, nanocrystalline

and ceramic samples of BaTiO3.39-46 However, the feature of this

spectrum is quite different from those of other materials measured

in our previous work, such as Ba(Mg1/3M2/3)O3 (M=Ta, Nb),12,13,17

MgTiO3,14,15 and Ba(Mg1/2W1/2)O3.16 In our previous work, the

Raman spectra had flat baselines and sharp peaks [their full width

at half maximum (FWHM) was less than 30 cm- 1], while the

spectrum of BaTiO3 has uneven baseline and broad peaks (some

peaks have FWHM larger than 100 cm-1). The soft mode E(1) of

the tetragonal BaTiO3 may affect the baseline of the spectrum,23

and in this non-centrosymmetric space group, the polar modes (A1

and E) can be split into transverse optical (TO) modes and lon-

gitudinal optical (LO) mode,47 which may be the reason to widen

the peaks.

The Lorentz fittings are adopted to extract the peak frequencies

and the related FWHMs in the Raman spectrum, and the fitting

results are listed in Table 2. For achieving the best fitting, the

Lorentzians at 554.6 cm-1 was set to mimic the asymmetric shape

of the peak by consulting Venkateswaran's work.47 Since our te-

tragonal BaTiO3 sample is ceramic disk, the anisotropy of the

Raman modes does not perform, as the results, each Raman peak

may contain more than one vibrational mode. In this work, only

6 Raman peaks were obtained by our Lorentz fittings, although

the tetragonal BaTiO3 has 8 Raman active modes. Due to lack of

spectrum data at low frequencies (<100 cm-1), the frequency

position (~88 cm-1) for the soft mode E(1) may have relatively large

error. Since the peak shape of the Raman spectrum is affected by

various factors, such as defects, impurities, temperature, pressure,

grain size and so on, the Lorentz fitting results in Table 2 may

have some deviations from other literature.

According to our previous work,12-17 the vibrational modes

obtained by the DFT calculations corresponded to the related TO

modes in the vibrational spectra and the mode assignments were

mainly based on the mode frequency sequence given by the DFT

calculations. However, due to the presence of the soft mode, the

mode sequence at low frequencies for BaTiO3 is not accurate. For

example, the lowest frequency obtained by our DFT calculation

is 122.9 cm-1 corresponding to the A1 mode, while in the reference,24

the mode with the lowest frequency (36 cm- 1) showed E sym-

metry. Also, a more accurate computational result from the work

Fig.4 Sketches of the optical vibrational modes (color online)

Fig.5 Raman spectrum of BaTiO3 in the range of

70 to 1000 cm-1 (color online)The assignments for the vibrational modes are indicated.

1064

AN Wei et al.: Assignment for Vibrational Spectra of BaTiO3 Ferroelectric CeramicNo.6

of Evarestov and Bandura31 indeed shows an imaginary frequency.

Compared our DFT results with the Ref.31, though the detailed

value is different for mode frequencies, the sequence of the mode

frequencies is consistent. Additionally, for the tetragonal BaTiO3,

the splitting of the TO and LO modes makes the assignments

more difficult. DiDomenico et al.24 have measured the polarized

Raman spectra of the single-domain tetragonal BaTiO3 on dif-

ferent directions, and based on these polarized spectra, they have

given quite reliable symmetric assignments to the Raman modes

of the tetragonal BaTiO3. Therefore, we assigned all the six

Raman peaks based on the calculated mode frequency under the

restriction of the mode symmetry given in DiDomenico's polar-

ized spectra, and the results are represented in Table 2 and labeled

in Fig.5, which are in a reasonable agreement with the literature

reported assignments.24,39-46

According to the assignments, we can access the knowledge

that how the atoms vibrate in the real space for each mode. The

Raman-active-only mode B1 and nearly IR silent mode E(3), which

have some similarity on vibrations, are assigned to the sharp peak

at ~306.6 cm-1. Considering the O6 octahedron, the non-degen-

eracy B1 mode can be mainly described as the O4 atoms vibrating

against the O5 atoms along z axis or the wing-flapping vibration

of the O4-O5 plane perpendicular to z axis. Since this mode shows

no obvious change of the dipole moment, it is a non IR active

mode. While the double-degeneracy E(3) mode mainly refers to the

O3 atoms vibrating against the O4 atoms along x axis or against

the O5 atoms along y axis and they also can be described as the

wing-flapping vibration of the O3-O4 plane perpendicular to x

axis or that of the O3-O4 plane perpendicular to y axis. This mode

involves only slight change of the dipole moment, thus it results

in a nearly silent IR mode (see the data in Table 3). The high

frequency modes A1(3) and E(4) are assigned to the peak at 516.3 cm-1,

both of which also have some similarity on vibrations. The A1(3)

mode can be described as the vibration of the O3-O3 axis against

the O4-O5 plane along z axis in the O6 octahedron; while the E(4)

mode can be described as the vibration of the O5-O5 axis against

the O3-O4 plane along x axis or the O4-O4 axis against the O3-

O5 plane along y axis.

Except the B1 mode, all the other modes for the tetragonal

BaTiO3 are IR active ones, thus the FIR reflective spectrum is a

complement of Raman spectrum for further understanding the

vibrational modes of BaTiO3. The FIR reflective spectrum we

measured is shown in Fig.6(a): the main panel shows the FIR

details in the range from 100 to 1200 cm-1 and the inset shows the

data in the mid-infrared range from 500 to 4000 cm-1, respec-

tively. The profile of our IR reflective spectrum agrees well with

the data in literature for polycrystalline samples.25,48,49 Moreover,

the main feature of the spectrum for single crystal samples for the

tetragonal BaTiO3, no matter whether they are polarized in fer-

roelectric axis or not, can be captured from that for our powder

sample.23,50 Taking into account of the TO and LO splitting, the

FIR reflective spectrum for the tetragonal BaTiO3 is contributed

by 3A1(TO)+3A1(LO) and 4E(TO)+4E(LO).

By using the FOCUS program, we fit the FIR reflective

spectrum based on the FPSQ model.5 In this FPSQ model, the

complex dielectric function is described as:

ε*(ω) = εʹ(ω)- iεʺ(ω) = ε∞∏j = 1

n Ω 2jLO -ω

2 + iωγ jLO

Ω 2jTO -ω

2 + iωγ jTO

(6)

where ε¥ denotes the optical permittivity, n is the total number of

the IR active vibrational modes, ΩjTO, γjTO and ΩjLO, γjLO represent the

frequencies and damping factors of the jth transverse and longi-

tude modes of the vibrations, respectively, ω is frequency. The

complex permittivity ε* is related to the IR reflectivity (R), based

on the following Fresnel equation:

R =|

|||

|

|||ε* - 1

ε* + 1

2

(7)

Since the peaks related to the A1 and E modes are overlapped in

the spectrum for our ceramic sample, the FPSQ fitting is quite

cumbersome. It was mentioned above that all the A1 modes vibrate

Table 2 Parameters of the Raman active modes

No.

1

2

3

4

5

6

Mode (cal.)

E(1)

A1(1)

E(2)

A1(2)

B1

E(3)

A1(3)

E(4)

-

Ωcal/cm-1

123.2

122.9

169.9

169.8

256.0

256.3

503.0

503.0

-

Ωexp/cm-1

88.8

143.9

262.7

306.6

516.3

(554.6)

717.8

FWHMexp/cm-1

48.2

118.9

128.2

6.9

37.1

(86.5)

58.3

Mode47

E(1)(TO)

A1(1)(TO)

E(1)(LO)

E(2)(TO)

A1(1)(LO)

A1(2)(TO)

E(2)(LO)

E(3)(TO)

B1

E(4)(TO)

A1(3)(TO)

E(4)(LO)

A1(3)(LO)

Ωexp/cm-1 47

36

170

180

180

185

270

305

305

305

518

520

715

720

Table 3 Parameters of the IR active modes derived by the four-parameter model fitting

ε¥(xy)=2.0, ε¥(z)=7.7

No.

1

2

3

4

5

6

7

Mode (cal.)

E(1)

A1(1)

E (2)

A1(2)

E (3)

E (4)

A1(3)

Ωcal/cm-1

123.2

122.9

169.9

169.8

256.3

503.0

503.0

Intensity/(km∙mol-1)

3104.8

3100.8

15.3

17.5

0.0

587.0

586.4

ΩjTO/cm-1

24.0

177.6

181.2

279.5

308.4

482.5

497.1

γjTO /cm-1

101.1

0.4

6.6

143.7

7.9

19.4

46.0

ΩjLO/cm-1

178.8

191.7

308.2

464.2

463.5

686.8

708.4

γjLO/cm-1

7.6

30.3

8.1

21.0

5.9

83.5

5.1

εj

1451.6

11.0

0.6

30.2

0.0

0.2

1.5

1016(tanδj/ω)/Hz-1

58126.5

0.9

0.0

366.3

0.0

0.0

1.8

1065

Acta Phys. -Chim. Sin. 2015 Vol.31

along z axis and E modes are on xy-plane, thus the total reflec-

tivity (R) can be separated into the reflectivity on xy plane (Rxy)

and that along z axis (Rz):25

R(ω)=(2/3)Rxy(ω)+(1/3)Rz(ω) (8)

Both Rxy and Rz can be calculated using Eq.(7) independently, as

been done for the single crystal sample in literature.50 Therefore,

the overlapped A1 and E modes can be fitted separately.

The parameters of all the IR modes extracted from the best

fitting of the FIR reflective spectrum are listed in Table 3, which

are in a good agreement with the literature data for single crystal

tetragonal BaTiO3.50 In addition, the real and imaginary permit-

tivities (ε' and ε") were calculated based on the mode parameters

and shown in Fig.6(b) and Fig.6(c). According to the mode fre-

quencies and intensities obtained by the DFT calculations and the

mode symmetry deduced from our FIR spectrum, the IR spectrum

was assigned. The assignments are listed in Table 3 and also la-

beled in Fig.6, which are well consistent with the Raman spectrum

assignments shown in Table 2 and Fig.5. Even the FWMHs,

which are the representations of the damping factors, are in a good

agreement with the IR and Raman data. Moreover, the assignment

based on the IR spectrum has better resolution than that of the

Raman one for the overlapped peaks. The assignments in this

work are also well consistent with reported work.25,48,49

The contributions of each IR active mode to the dielectric

properties (permittivity εj and dielectric loss tanδj/ω) in the mi-

crowave frequency range can be calculated by the following

equations:51,52

ε j =ε∞Ω 2

jTO

∏k

(Ω 2kLO -Ω

2jTO)

∏k ≠ j

(Ω 2kTO -Ω

2jTO)

(9)

tan δ j /ω =ε jγ jTO/Ω 2

jTO

ε∞ +∑j

ε j

(10)

The calculated results are listed in Table 3. The data indicate

that the E(1) mode at the lowest frequency has the largest contri-

butions to both εj and tanδj/ω, several magnitudes larger than other

modes. Except the abnormal E(1) mode, εj and tanδj/ω for the other

modes have the similar values to most ceramic material. Com-

paratively, the A1(1) mode at 177.6 cm-1 and the A1

(2) mode at 279.5

cm-1 have relatively large values of εj, to all the rest modes.

As shown in Fig.4, the soft E(1) mode can be described as the

vibration of the O6 octahedral cage against the inner Ti atom al-

most along x or y axis in the TiO6 octahedron, which gives rise to

a significant dipole moment change and leads to huge dielectric

permittivity εj. Quite similar to E(1) mode, the A1(1) mode can also be

considered as the vibration of the O6 octahedral cage against the

inner Ti atom, but it is along z axis instead of xy plane. The vi-

bration of the A1(2) mode, another large contributor to εj and tanδj/ω,

can be viewed as the vibration of the TiO6 octahedron against Ba

atom along z axis. The E(2) mode, a less contributor to the di-

electric properties, has similarly vibrational pattern to the A1(2)

mode, but the vibration of the TiO6 octahedron against Ba atom on

xy plane.

The dielectric property parameters, such as the optical per-

mittivity ε¥, the static permittivity ε, and the quality factor Q´f can

be obtained not only from the FPSQ fittings, but also from the

DFT calculations, and the microwave or the AC impedance

measurements, which are comparatively listed in Table 4. By the

FPSQ fitting data, ε¥ was calculated from ε¥(xy) and ε¥(z) following

the equation (11), ε and Q´f of BaTiO3 were calculated by the

equations (12, 13), respectively.

ε∞ = 23ε∞(xy) + 1

3ε∞(z) (11)

ε = ε∞ +∑j

n

ε j (12)

Q × f = 1/∑j

n

(tan δ j /ω) (13)

For the DFT first-principles calculation, the values of ε¥ and ε

were obtained based on pure mechanical phonon information

which contains no splitting of TO and LO. Due to the large di-

Fig.6 Fitting results for the FIR reflective spectrum of BaTiO3

using the four-parameter model (a), ε' spectrum (b), and ε''

spectrum (c) (color online)ε': real permittivity; ε'': imaginary permittivity

Table 4 Dielectric property parameters deduced

from the various methods

calculation

FPSQ fitting

measurement

ε¥

6.8

3.9

-

ε

169.1

1499.0

1513

(Q´f)/GHz

-

171 (at 1 GHz)

86 (at 1 MHz)

1066

AN Wei et al.: Assignment for Vibrational Spectra of BaTiO3 Ferroelectric CeramicNo.6

electric loss, the microwave measurements were failed because of

absence of the resonance peak. The measured dielectric param-

eters listed in Table 4 were obtained by the AC impedance mea-

surements.

The data in Table 4 indicates that the values of the permittivity

obtained by the FPSQ fitting and AC measurement are in a good

agreement with each other. The permittivity of the tetragonal

BaTiO3 is much larger than that of non- ferroelectric materials

(usually it is less than 200), thus BaTiO3 is considered as the

ferroelectrics. The large permittivity is mainly contributed by the

E(1) soft mode, and the A1(1) mode also has quite large contribution.

These two modes actually are degenerated from the F1u soft mode

of the high temperature cubic BaTiO3, which controls the cubic-

tetragonal phase transition of BaTiO3. With a decrease of tem-

perature to the phase transition temperature (or Curie tempera-

ture), the vibration of the triple degeneracy F1u mode along z axis

is partially frozen and the mode is separated to the A1(1) mode and

the E(1) mode, which cause the ferroelectric features of BaTiO3: (i)

the appearance of the A1(1) mode leads to the crystal polarization

along the c direction (z axis); (ii) the E(1) mode keeps its soft mode

behavior corresponding to the large permittivity on the a or b

direction (x or y axis). These features are consistent with the di-

electric properties measured for the single crystal sample of Ba-

TiO3 in early time.50,53 The permittivity obtained by the DFT cal-

culation is clearly larger than the values calculated for the non-

ferroelectric materials (<100),13,14,16,17 which indicates that the DFT

calculations can reveal the ferroelectric characters of the mate-

rials. However, the calculated permittivity is much less than the

FPSQ fitting value and the AC measured value. As we know, the

large permittivity is mainly contributed by the E(1) soft mode, it is

thought that the error is large for the CASTEP package to deal

with the soft modes. Since the microwave measurements (in GHz)

are not applied, the data measured in MHz are listed in Table 4.

For the Q´f, the FPSQ fitting gave the value at GHz, which is

related to the anharmonic vibration of the lattice; while the AC

measurement gave the value at MHz, which corresponds to

conductance of charge carriers.53,54 We cannot make comparison

between the two Q´f values in Table 4, but it can be seen that

these two values are much smaller than those for the non-ferro-

electric materials (>105 GHz) we previously studied.12-17 Although

the ferroelectric materials have an advantage of large permittivity,

they also have a disadvantage of large dielectric loss.

4 ConclusionsBaTiO3 ceramic disk was synthesized and sintered by using

conventional solid state reactions at 1400 °C for 4 h. The pure

tetragonal phase of the disk was confirmed by the XRD data re-

finement. Based on the group theory analysis, the symmetry co-

ordinates of the tetragonal BaTiO3 were obtained and all the vi-

brational modes were presented by the linear combinations of

symmetry coordinates. The A1 and E modes are both Raman and

IR active, while the B1 mode is Raman active only. The Raman

and the FIR reflective spectra measured in this work were fitted

by Lorentz function and the FPSQ model, respectively, and the

assignments of these vibrational spectra were conducted based on

the DFT calculations and with the assistance of the literature data.

For the assignments, the TO and LO splittings of the polar modes

were considered. The E(1) soft mode at the lowest frequency can

be described as the vibration of the Ti atom against O6 octahedral

cage in the TiO6 octahedron on xy plane, which is the most con-

tributing mode to the large permittivity and dielectric loss. The A1(1)

mode has the similar vibrational feature to the E(1) mode, but it is

along z axis, and it is correlated to the crystal polarization along

z axis. Raman-active-only B1 mode can be viewed as the wing-

flapping vibration of the O4-O5 plane perpendicular to z axis in

the O6 octahedron. Both the A1(1) and E(1) modes are degenerated

from the triple degeneracy F1u mode of the cubic BaTiO3. Though

the DFT calculation causes some errors for dealing with soft

mode, it still serves a good reference to understand the characters

of the vibrational modes and the mechanism of dielectric prop-

erties of the materials.

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