Nghiên cứu nâng cao tính chất sắt điện của màng mỏng PZT chế tạo bằng công nghệ sol gel

MINISTRY OF EDUCATION AND TRAINING
HANOI UNIVERSITY OF TECHNOLOGY
INTERNATIONAL TRAINING INSTITUTE FOR MATERIALS SCIENCE
..

LE VAN MINH

IMPROVEMENT IN FERROELECTRIC PROPERTY OF PZT THIN
FILMS FABRICATED BY SOL-GEL METHOD

MASTER THESIS OF MATERIALS SCIENCE
BATCH ITIMS - 2006

Supervisor: Associate Prof. Dr. VU NGOC HUNG

Hanoi - 2008


BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC BÁCH KHOA HÀ NỘI
VIỆN ĐÀO TẠO QUỐC TẾ VỀ KHOA HỌC VẬT LIỆU

LÊ VĂN MINH

NGHIÊN CỨU NÂNG CAO TÍNH CHẤT SẮT ĐIỆN CỦA
MÀNG MỎNG PZT CHẾ TẠO BẰNG CÔNG NGHỆ SOL-GEL

LUẬN VĂN THẠC SĨ KHOA HỌC VẬT LIỆU
KHOÁ ITIMS - 2006

Người hướng dẫn khoa học: PGS. TS. VŨ NGỌC HÙNG

Hà Nội - 2008


ACKNOWLEDGMENTS
My most sincere gratitude belongs to Associate Professor Vu Ngoc Hung
for his continuous support and advice in my research efforts. Moreover, besides
expanding my breadth of ferroelectric knowledge, Prof. Hung has made working
at MEMS Group, International Training Institute for Materials Science (ITIMS),
a true pleasure.
I’d also like to acknowledge IMS, Twente University, MESA+, the
Netherlands, for analyzing and measuring experiment samples including XRD,
AFM, FE-SEM, and TF2000 piezoelectric mode. Special thanks go to PhD.
Nguyen Duc Minh, IMS, Twente, University, MESA+, the Netherlands for
directly implementing these analyses and measurements.
I would also like to express my appreciation to the rest of the MEMS
Group including, Dr. Trinh Quang Thong, Nguyen Van Minh for their daily help
and sincere friendship. Special thanks go to Ms. Bui Thi Huyen, an
undergraduate student, for spending her time to take experiments along with me
and Ms. Vu Thu Hien, who investigated PZT thin film materials in 2007 and
helped establish the scientific kernel and intrigue for completing this current
work.
I would also like to thank ITIMS for making a scientific environment and
facilities. Special thanks go to Dr. Nguyen Anh Tuan and Ms. Le Thanh Hung
for sputtering supports and Dr. Pham Thanh Huy who gave me authority to use
his group’s furnace system.
Finally, I would like to thank my family, especially Mom, Dad, my old
brothers, Le Van Son and Le Van Quang, for their continual love and support as
well as their forbearance of my scientific babble. Without this link to family, I
would become totally consumed by science, which would be a sad life indeed.

Hanoi, September, 2008
LE VAN MINH


VIỆN ĐÀO TẠO QUỐC TẾ VỀ KHOA HỌC VẬT LIỆU
ITIMS KHỐ 2006
Tiêu đề luận văn:
“Nghiên cứu nâng cao tính chất sắt điện của màng mỏng PZT
chế tạo bằng công nghệ sol-gel”
Tác giả:
Lê Văn Minh
Người hướng dẫn:
PGS.TS. Vũ Ngọc Hùng
Người nhận xét: 1.
2.

Tóm tắt:
Màng mỏng PZT [(PbZrxTi1-x)O3], với các tính chất sắt điện, áp điện, hoả
điện nổi bật là một trong những vật liệu được nghiên cứu và ứng dụng trong lĩnh
vực cơng nghệ MEMS, bộ nhớ FeRAM, vv… Tại phịng thí nghiệm MEMS của
Viện ITIMS, màng mỏng PZT cũng đã được triển khai nghiên cứu trong thời gian
vài năm trở lại đây. Bước đầu đã xây dựng được quy trình tổng hợp sol-gel và chế
tạo màng mỏng PZT, xong còn một số hạn chế về tính chất của màng như màng rạn
nứt, nhiều rosettes, độ phân cực điện dư (Pr) nhỏ và điện trường khử phân cực (Ec)
lớn. Mục tiêu của luận văn này là tiến hành những nghiên cứu tiếp theo để tạo ra
màng mỏng PZT có tính chất nâng cao nhằm định hướng trong các ứng dụng của
nhóm MEMS. Trong luận văn này chúng tôi tập trung nghiên cứu nhằm nâng cao
tính chất sắt điện, tránh việc nứt màng và giảm thiểu rosettes của màng mỏng PZT.
Bằng việc tối ưu hóa quy trình sử lý nhiệt, nghiên cứu ảnh hưởng của vật liệu điện
cực đế, và đặc biệt là nghiên cứu màng mỏng PZT dị lớp, tính chất của màng đã

được nâng cao rõ rệt. Màng mỏng PZT(53/47) trên đế Pt(111)/Ti/SiO2/Si có định
hướng ưu tiên (100) đã được chế tạo thành cơng có các thơng số Pr và Ec đạt các giá
trị tương ứng là 12µC/cm2 và Ec là 80kV/cm. Màng PZT(53/47) trên đế oxide
SRO(110)/YSZ/Si có giá trị Pr và Ec tương ứng là 24µC/cm2 và 65kV/cm. Màng
PZT(47/53)/PZT(53/47) cấu trúc dị lớp trên đế SRO/YSZ/Si đã đạt được giá trị Pr
và Ec tương ứng là 28µC/cm2 và 60kV/cm.
Từ khoá: Màng mỏng PZT; Sol-gel; Màng mỏng sắt điện; PZT cấu trúc dị lớp;
Điện cực đế SRO; Điện cực đế Platinum.


INTERNATIONAL TRAINING INSTITUTE FOR MATERIALS SCIENCE
BATCH -2006
Title of Msc. Thesis:
“Improvement in Ferroelectric Property of PZT Thin Films
Fabricated by Sol-gel Method”
Author:
Le Van Minh
Supervisor:
Associate Prof. Dr. Vu Ngoc Hung
Referees:
1.
2.

Abstract:
PZT [(PbZrxTi1-x)O3]thin films having prominent ferroelectric, piezoelectric,
and pyroelectric properties have researched and applied in numerous technological
fields, such as MEMS technology and nonvolatile FeRAM and so on. In recent
years, PZT thin films have been studied and fabricated at MEMS Group,
International Training Institute for Materials Science (ITIMS). At the first stage, the
PZT thin films were synthesized and fabricated. There were, however, some

disadvantages of these thin films, such as, cracked films, many rosettes, small
remanent polarization Pr, and high coercive field Ec. This thesis is the next study
with the aim of fabricating high quality PZT thin films for oriented applications of
MEMS Group. Thus, eliminating crack, reducing rosettes, and improving
ferroelectric property were studied. By optimizing the thermal process and studying
the influence of electrode materials and the heterolayered PZT thin films, the
properties of the PZT thin films have improved significantly. The PZT thin films on
Pt(111)/Ti/SiO2/Si substrates had highly preferred-(001) orientation and the values
of remanent polarization and coercive filed were 12µC/cm2 and 80kV/cm,
respectively. The remanent polarization and the coercive field of the PZT thin films
on SRO(110)/YSZ/Si substrates were 24µC/cm2 and 60kV/cm, respectively. These
values of the heterolayered PZT(47/53)/PZT(53/47) thin films with SRO bottom
electrodes were 28µC/cm2 and 60kV/cm, respectively.
Keywords: PZT thin films; Sol-gel method; Ferroelectric thin films; Heterolayered
PZT thin films; SRO electrode; Platinum electrode.


i

CONTENTS

List of Tables

iv

List of Figures

v

Abbreviations

x

Chapter 1:

FUNDAMENTAL THEORIES OF PZT THIN FILM

1

1.1 Description of Ferroelectricity

1

1.1.1

Description of the Ferroelectric Phase Transition

3

1.1.2

Ferroelectricity in Perovskite Crystals

4

1.1.3

Origin of Spontaneous Polarization

5

1.2 Ferroelectric Domains

7

1.2.1

Domain Walls

1.2.2

Field-Induced Strain Mechanism and Domain Configurations

1.2.3

9

in Ferroelectric Ceramics

10

Extrinsic Contribution to Ferroelectric Properties

12

1.3 PZT thin films

14

1.3.1

Effects of Stress in PZT thin films

16

1.3.2

Effects of Crystalline Orientation on PZT thin films

17

1.4 Ferroelectric Aging

18

1.5 Ferroelectric Fatigue

21

1.5.1

General Discussion of Fatigue Mechanisms

22

1.5.2

Fatigue – Domain Wall Pinning Hypothesis

23

1.5.3

Fatigue- Seed Inhibition Hypothesis

24


ii

1.6 Ferroelectric thin film applications
References:
Chapter 2:

28
31

EXPERIMENTS

36

2.1 General PZT thin film fabrication methods

36

2.1.1

Sputtering

36

2.1.2

PLD: Pulsed Laser Ablation

37

2.1.3

Sol-gel: Solution-Gel

37

2.2 The Sol-Gel Process

39

2.2.1

Sol-gel Route

39

2.2.2

2-MOE based PZT solution preparation

41

2.2.3

Thermal analysis

43

2.3 PZT thin film Fabrication and Processing

46

2.3.1

Substrate Preparation

46

2.3.2

Deposition

50

2.3.3

Thermal Processing

52

References:
Chapter 3:

54
RESULTS AND DISCUSSION

3.1 PZT (53/47) thin films on Pt/Ti/SiO2/Si substrates

55
55

3.1.1

Effect of thermal process

55

3.1.2

Microstructure and XRD patterns

57

3.1.3

Electrical Properties

61

3.2 PZT(53/47) thin films on SRO bottom electrode
3.2.1

Microstructure and XRD patterns

64
66


iii

3.2.2

Electrical properties of PZT(53/47) on SRO/YSZ/Si substrate 68

3.2.3

C-V characteristics

3.3 Heterolayered PZT(47/53)/PZT(53/47) Thin Films

70
71

References:

76

CONCLUSIONS

78


iv

List of Tables
Table 2.1:

Advantages and disadvantages of different deposition
techniques

Table 2.2:

39

Recipe for 0.01mol 2-methoxyethanol (2-MOE) based
PZT (53/47) solution (0.4M).

41

Table 2.2:

Flow diagram for the synthesis of the PZT solution[11-12] 43

Table 3.1:

Remnant polarization and coercive field of PZT thin films
on Pt/Ti/SiO2/Si substrates under different thermal
processes

62


v

List of Figures
Figure 1-1:

Typical hysteresis loops from various ferroelectric
ceramics: paraelectric (a), ferroelectric (b), relaxor (c),
antiferroelectric (d)

Figure 1-2:

2

Soft Mode Description of the Ferroelectric Phase
Transition

3

Figure 1-3:

Illustration of the Perovskite Unit Cell

5

Figure 1-4:

Energy

explanation

of

the

origin

of

spontaneous

polarization

6

Figure 1-5:

Formation of 180o Ferroelectric Domain Walls

9

Figure 1-6:

Strain Energy Relief by 90o Domain Wall Motion

Figure 1-7:

Schematic explanation of the strain change in a
ferroelectric

ceramic

associated

with

the

10

domain

reorientation [11].
Figure 1-8:

Illustration of Dielectric Tunability in Ferroelectric
Materials

Figure 1-9:

13

Phase Diagram of the PbZrO3/PbTiO3 Solid Solution
System

Figure 1-10:

13

15

Illustration of stress induced domain structure in PZT thin

films

17

Figure 1-11:

Aging in Poled Ferroelectric Materials

20

Figure 1-12:

Effect of Aging in Unpoled Ferroelectrics

20

Figure 1-13:

Change in P-E Response as a result of ferroelectric fatigue 21

Figure 1-14:

Domain Wall Pinning Model of Ferroelectric Fatigue

24


vi

Figure 1-15:

Seed Inhibition Model of Ferroelectric Fatigue

Figure 1-16:

AFM assisted Piezo-Response Images of a Fatigued PZT

25

film

26

Figure 1-17:

Overview of typical ferroelectric thin-film applications

28

Figure 1-18:

Example of a (a) sensor and (b) an actuator. A PZT layer is
shown in these examples but it could be a different
piezoelectric layer. The microvalve in (b) is based on
reference [42].

29

Figure 2-1:

Sol-gel System at MEMS Group, ITIMS.

42

Figure 2-2:

DSC and TGA analysis of PZT 2-MOE based sol-gels.

44

Figure 2-3:

Four regions of solid appear as the calcining temperature
increases

Figure 2-4:

45

Schematic showing boundary of stability of a range of
metals with respect to their oxides [5].

48

Figure 2-5:

Unit cell of YSZ material [5]

49

Figure 2-6:

The structure of SRO/YSZ/Si substrate [6]

49

Figure 2-7:

Stages of the spinning process (T0
50

Figure 2-8:

Spinner for depositing PZT solution onto substrates:
Spinner (a), and diagram of spinning rate(b)

51

Figure 2-9:

Flow diagram of Thermal Process

53

Figure 3-1:

Thermal process 1(a): one step of pyrolysis at 300oC for
1min.; thermal process 2(b): two steps of pyrolysis at
300oC for 30 min. and 400oC for 10 min.

55


vii

Figure 3-2:

Micrographs of the PZT thin film fabricated by using
thermal process 1, two-layer thin film (a), and four-layer
thin film (b)

Figure 3-3:

Micrograph of four-layer PZT(53/47) on Pt/Ti/SiO2/Si
fabricated by using the new thermal process

Figure 3-4:

58

The XRD patterns of the PZT films with different mole
excess Pb (a) 5%, (b) 10%, and (c) 20% [15]

Figure 3-7:

58

Cross section SEM image of PZT(53/47) thin film on
Pt/Ti/SiO2/Si

Figure 3-6:

57

Atomic Force Microscopy image of PZT 53/47 thin film
fabricated by using the thermal process 2

Figure 3-5:

56

59

The XRD pattern of the PZT (53/47) thin films on
Pt/Ti/SiO2/Si fabricated by using the thermal process 1(a)
and the new thermal process 2(b).

Figure 3-8:

60

Ferroelectric hysteresis loops of PZT thin films treated by
thermal process 1: (a) at different frequencies (b) at
frequency 1kHz with different applied voltages 5V, 10V,
and 20V [1]

Figure 3-9:

61

Ferroelectric hysteresis loops of PZT thin films treated by
thermal process 2: (a) at different frequencies (b) at
frequency 1kHz with different applied voltages 5V, 10V,
and 20V

Figure 3-10:

62

The capacitance (a) and dielectric constant (b) of PZT thin
films versus applied voltages

63


viii

Figure 3-11:

The

P-E

and

C-V

characteristics

of

PZT(53/47)/Pt/Ti/SiO2/Si
Figure 3-12:

Micrograph of PZT thin films on the SRO/YSZ/Si
substrates

Figure 3-13:

67

XRD pattern of four-layer PZT(53/47) thin film on
SRO/YSZ/Si substrate

Figure 3-16:

66

SEM of four-layer PZT(53/47) thin film on SRO/YSZ/Si
substrate

Figure 3-15:

65

AFM images of four-layer PZT (53/47)thin film on
SRO/YSZ/Si substrate

Figure 3-14:

64

67

Ferroelectric hysteresis loops of PZT/SRO/YSZ/Si thin
films: (a) at different frequencies (b) at different applied
voltages 5V, 10V, and 20V with a frequency 1kHz

Figure 3-17:

P-E loops of PZT on SRO/YSZ/Si and Pt/Ti/SiO2/Si
substrates, at applied voltage 20V, and frequency 1kHz

Figure 3-18:

69

69

C-V and P-E characteristics of PZT (53/47) thin film on
SRO/YSZ/Si substrate at frequency 10kHz and at voltage
10V

Figure 3-19:

The dielectric constant of PZT thin films on SRO and Pt
bottom electrodes depend on applied voltages

Figure 3-20:

71

Illustrate the heterolayers between PZT(47/53)(tetragonal)
and PZT(53/47) (rhombohedral)

Figure 3-21:

70

Cross section SEM images of PZT thin film on
SRO/YSZ/Si substrates: four-layer PZT (53/47) (a) and

73


ix

four-layer PZT(47/53)/PZT( 53/47) heterolayer structure
(b)
Figure 3-22:

AFM images of heterolayered PZT thin film consisting of
alternating PZT(47/53) and PZT(53/47)layers

Figure 3-23:

73

74

P-E hysteresis for PZT thin film on SRO/YSZ/Si
substrates: PZT(53/47) (a), PZT(47/53)/PZT(53/47) (b)

75


x

Abbreviations
AFM
FeRAM
FE-SEM
MEMS
MPB
PZT
XRD

Atomic Force Microscopy
Ferroelectric Random Access Memory
Field Emission – Scanning Electron Microscopy
Micro-Electromechanical Systems
Morphotropic Phase Boundary
Lead Zirconate Titanate (PbZrxTi1-xO3)
X-ray Diffraction


1

Chapter 1:

1.1

FUNDAMENTAL THEORIES OF PZT THIN FILM

Description of Ferroelectricity
Ferroelectric materials are a subclass of pyroelectric crystals, while

pyroelectric crystals are a special class of piezoelectric crystals. Ferroelectrics
are commonly described as non-centrosymmetric materials containing an
electrically re-orientable spontaneous polarization at equilibrium. The
prominent properties of these materials are piezoelectricity, pyroelectricity,
and ferroelectricity.
Piezoelectric effect was discovered by Curie brother in 1880.
Piezoelectricity is a material property that linearly relates applied stresses to
induced dielectric displacements (direct piezoelectric effect) or applied
electric fields to induced strains (converse piezoelectric effect)[1]. A third
rank tensor is necessary to fully describe the piezoelectric response of a
crystal. Neumann’s principle establishes that the symmetry of a crystal’s point
group is reflected in the symmetry of its external properties. Thus, only
materials that lack an inversion point (i.e. are non-centrosymmetric) can
exhibit piezoelectricity. Centrosymmetric materials are incapable of
containing odd-rank tensor properties because the inversion symmetry
produces perfect compensation for such “directional” properties. Of the 32
point groups, 21 lack inversion point symmetry and 20 of these display
piezoelectricity (only the point group 432 fails to be piezoelectric) [2]. Thus,
the non-centrosymmetric crystalline establishes the piezoelectric nature of the
ferroelectric materials.
However, the piezoelectric response does not guarantee the “polar” nature
of a material. In some piezoelectric crystals such as quartz, polar directions
are arranged such that they self-compensate and only exhibit the piezoelectric

2

response under inhomogeneous stresses- not under hydrostatic conditions.
Only 10 of the 20 piezoelectric point groups (those that contain a unique axis
of rotation with no mirror plane perpendicular to it [3]) allow for the existence
of permanent dipoles- that is a unique polar axis [4]. These “polar” materials
will exhibit a hydrostatic piezoelectric response as well as the additional
property of pyroelectricity. In such crystals, temperature changes cause
thermal fluctuations in the basic ionic and electronic forces of the material,
varying the magnitude of the permanent dipole moments within the material.
The pyroelectric effect manifests as a variation in the polarization magnitude
as a result of a temperature change [4]. Pyroelectricity is intrinsically
established in ferroelectrics because of their “spontaneous” polarization.

Figure 1-1: Typical hysteresis loops from various ferroelectric ceramics:
paraelectric (a), ferroelectric (b), relaxor (c), antiferroelectric (d)
Yet, materials with a unique polar axis, such as ZnO or GaN, are not
necessarily ferroelectric. The final requirement, a switchable or “reorientable” dipole, is required for the ferroelectric response. Ferroelectricity is
essentially a hysteretic polarizability in response to an applied electric field.


3

As shown in figure 1-1b a plot of polarization vs. electric field has the classic
hysteresis loop shape analogous to the magnetization response of a
ferromagnet. P-E loops can be used as a fingerprint to identify the material.
The typical hysteresis loops obtained for various ferroelectric ceramics are
presented in figure 1-1.

1.1.1 Description of the Ferroelectric Phase Transition
Many approaches can be taken to understand the ferroelectric phase
transition; however, an examination of lattice dynamics is perhaps the most
useful for visualizing the structure-property relationships that occur in the
material. From this viewpoint, a displacive ferroelectric transition occurs if a
transverse optic branch of the phonon spectrum has an instability at the Curie

Figure 1-2: Soft Mode Description of the Ferroelectric Phase Transition:
Dispersion spectrum showing the change in a soft mode phonon when a
material is cooled below the Curie temperature. The “frozen” phonon
vibration (ω=0) in ferroelectric materials creates a set of distortion/dipoles
that oscillate with infinite wavelength (k=0) while anti-ferroelectric materials
have dipoles that oscillate with a wavelength that is twice the unit cell
dimension (k=1/λ=1/2a) [4].


4

temperature. As the instability is approached from high temperatures, this
phonon mode “softens” until its frequency reaches zero and the wavelength
becomes infinite. Thus, at the transition temperature, the soft mode vibration
locks in a displacive distortion which creates the spontaneous polarization
necessary for ferroelectricity. However, this vibrational mode remains easily
activated, providing a significantly large contribution to the dielectric
permittivity than other modes. Figure.1-2 demostrates how soft modes can
create both ferroelectric and anti-ferroelectric behavior (the latter being a nonpolar state with alternating opposite dipoles) [4, 5].
1.1.2 Ferroelectricity in Perovskite Crystals
Ferroelectrics with the perovskite crystal structure are probably the most
important class of ferroelectrics for technological applications. An ideal
perovskite structure is depicted in figure 1-3 and is described as having a

cubic unit cell with A-site cations at the corners, a B-site cation in the center,
and oxygen anions on the six centered faces. This structure is capable of
adjusting through minor distortions to accommodate A and B site cations with
inexact radii sizes to achieve ideal 12-fold and 6-fold respective
coordinations. In the ideal perovskite structure, twice the B-O bond length
(2rB-O)should equal the unit cell dimension (a), while twice the A-O bond
length (2rA-O) should equal the face diagonal (a√2). From this analysis, a
tolerance factor can be derived:

𝑡𝑡𝑓𝑓 =

√2𝑟𝑟𝐴𝐴−𝑂𝑂
2𝑟𝑟𝐵𝐵−𝑂𝑂

(1-1)

where the tolerance factor (tf) should equal 1 for the ideal case. When the Bsite cation is too small for its site (tf>1), such as in BaTiO3 and PbTiO3, this
cation shifts towards an oxygen face, taking on an effective 5-fold
coordination. To accommodate the reduction of the B-O bond length, the unit


5

cell contracts in the plane perpendicular to the shift creating a tetragonal
distortion. Conversely, if the A-site cations are too small (tf<1), the oxygen
octahedron tilts. The octahedron may tilt towards the corners to create a
rhombohedral distortion such as in Zr-rich PZT or towards the edges to create
an orthorhombic distortion such as in the antiferroelectric PbZrO3 [6]. Since
the space group of cubic perovskite (Pm3m) is centrosymmetric, these
distortions are necessary to remove the center of symmetry and stabilize the

existence of permanent dipoles.

Figure 1-3: Illustration of the Perovskite Unit Cell: A drawing of the
undistorted perovskite unit cell with the location of A-site and B-site cations
labeled
Although some technologically important ferroelectrics such as LiNbO3
take on layered structure, the remaining portions of this chapter will focus on
ferroelectricity in perovskite-based materials unless otherwise noted.
1.1.3 Origin of Spontaneous Polarization
In a polar structure, each dipole is influenced by its neighbors. For
simplicity, assuming the dipole moments results from the equal displacement


6

of one kind of ion A relative to the crystal lattice, in which exists a local field,
(Eloc) from the surrounding polarization P to any individual ion A.
𝐸𝐸𝑙𝑙𝑙𝑙𝑙𝑙 = (𝛾𝛾/3𝜀𝜀0 )𝑃𝑃

(1-2)

where 𝛾𝛾 is the Lorentz factor, 𝜀𝜀0 is the permittivity of vacuum.

This local field is the driving force of the ion shift. Thus, the Lorentz

factor is sometimes seen as a measure of the strength of dipole interactions. If
the ionic polarizability of ion is 𝛼𝛼, then the dipole moment of the unit cell is:
𝜇𝜇 = (𝛼𝛼𝛼𝛼/3𝜀𝜀0 )𝑃𝑃

(1-3)

Considering N number of atoms per unit volume, the energy of the dipole
moment is:
𝑊𝑊𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑁𝑁(−𝜇𝜇𝐸𝐸𝑙𝑙𝑙𝑙𝑙𝑙 ) = −(𝑁𝑁𝑁𝑁𝛾𝛾 2 /9𝜀𝜀02 )𝑃𝑃2

(1-4)

When the ions are displaced from their nonpolar equilibrium positions, the
increase of the elastic energy per unit volume is:
𝑊𝑊𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 𝑁𝑁[(𝑘𝑘/2)𝑢𝑢2 + (𝐾𝐾/4)𝑢𝑢4 ]

(1-5)

Where u is the ions displacement from their equilibrium positions (u=P/Nq,
where q is the electric charge), k and K are force constants.
The total energy can be expressed as (Figure 1-4 [1]):
𝑘𝑘

𝑊𝑊𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑊𝑊𝑑𝑑𝑑𝑑𝑑𝑑 + 𝑊𝑊𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = �(2𝑁𝑁𝑞𝑞2) −

(a) Dipole Interaction

𝑁𝑁𝑁𝑁𝛾𝛾2
�9𝜀𝜀02 �

𝐾𝐾

� 𝑃𝑃2 + �(4𝑁𝑁3

(b) Elastic Energy

� 𝑃𝑃4

𝑞𝑞4 )

(1-6)

(c) Total Energy

Figure 1-4: Energy explanation of the origin of spontaneous polarization
(after K. Uchino [7])


7

The first order derivative of the total energy Wtot related to polarization P
shows:
𝜕𝜕𝑊𝑊𝑡𝑡𝑡𝑡𝑡𝑡

Hence,

𝜕𝜕𝜕𝜕

𝑘𝑘

= 𝑃𝑃 �(𝑁𝑁𝑞𝑞2) −
𝑃𝑃2 =

2𝑁𝑁𝑁𝑁𝛾𝛾2
�9𝜀𝜀02 �

𝐾𝐾

� + �(𝑁𝑁3

� 𝑃𝑃3 = 0

𝑞𝑞4 )

��2𝑁𝑁𝑁𝑁𝛾𝛾2 /9𝜀𝜀02 �−(𝑘𝑘/𝑁𝑁𝑞𝑞2 )�
𝐾𝐾/𝑁𝑁3 𝑞𝑞4

(1-7)

(1-8)

From Eq. (1-8), one can see that there are two solutions for P where the total
energy will be minimum. This has been represented in figure 1-4c.
1.2

Ferroelectric Domains
When a perovskite ferroelectric is cooled below Tc, the direction of the

spontaneous polarization is equally probable in all of the crystallographically
equivalent polarization directions – for instance, all six <100> directions for
tetragonal distortions or all eight <111> directions for rhombohedral
distortions. Inevitably, electrical and mechanical boundary conditions limit
the volume over which a particular polarization direction can extend. As a
result, regions of uniformly oriented spontaneous polarizations develop called
domains. Domain walls separate domains with different spontaneous

polarization orientation [5, 8].
Physically, the electrical constraint imposed on the material is a
depolarization field developed from a “surface charge” that results from the
emergence of the spontaneous polarization, because a discontinuity in the
periodic dipole alignment occurs at the surface or at an interface (grain
boundary), an electric field forms due to the uncompensated dipole charges at
this surface (“surface charge”). This field must act in opposition of the
dipoles. Mathematically this situation is expressed using the dielectric


8

displacement vector (D), which describes the surface charge density of the
material:
𝐷𝐷 = 𝜀𝜀0 𝐸𝐸 + 𝑃𝑃

(1-10)

where ε0 is the permittivity of free space (8.8510-12F/m), E is the applied
electric field, and P is the induced polarization of the material, which is also a
function of the electric field:
𝑃𝑃(𝐸𝐸 ) = 𝑃𝑃𝑠𝑠 + 𝜀𝜀0 𝜒𝜒𝜒𝜒

(1-11)

where Ps is the spontaneous polarization and χ is the dielectric susceptibility
[5, 9]. Assuming no charge compensation from outside charges (D=0) and
using the definition of the relative permittivity (dielectric constant) of a
material (εr=1+χ) these expressions become:
𝐸𝐸𝐷𝐷 = −

𝑃𝑃𝑠𝑠

(1-12)

𝜀𝜀0 𝜀𝜀𝑟𝑟

where ED is the depolarization field for spontaneous polarization. The volume
density of energy stored by this field is then given by:
𝑑𝑑𝑑𝑑 = ∫ 𝐸𝐸𝐷𝐷 . 𝑑𝑑𝑑𝑑 = ∫ −

𝑃𝑃𝑠𝑠

𝜀𝜀0 𝜀𝜀𝑟𝑟

𝑑𝑑𝑃𝑃

(1-13)

which implies that the energy associated with the depolarization field inside
the system increase with domain size as:
𝑈𝑈𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ∝

𝑃𝑃𝑠𝑠2

2𝜀𝜀0 𝜀𝜀𝑟𝑟

𝑉𝑉

(1-14)

Where Ufield is the energy and V is the volume of the domain [9]. At some
volume, the increase in energy will make a single domain unstable. To lower
its energy, the domain can split into two domains with different spontaneous
polarization directions. However, the domain wall that separates the two new
domains has an interfacial energy associated with it. The balance of
depolarization field energy and domain wall interfacial energy partially
determines domain size. A similar analysis can be undertaken for strainrelated energies imposed by the lattice distortions undergone at the phase


9

transition. This strain energy also influences domain size, specifically in
ceramic (polycrystalline) ferroelectrics [8, 9].
Immediately after cooling below Tc, the complex electrostatic and
elastic constraints present in a polycrystalline ceramic lead to a complex
domain structure that often exhibits not net polarization direction. Thus, the
ceramic is non-polar (neither piezoelectric nor pyroelectric) until the domains
are aligned with the application of an external electric field (poling) [5, 8].
Because ceramics have randomly oriented grains, the maximum spontaneous
polarization is a fraction of the single-crystal value and has been calculated to
be 0.83, 0.87, and 0.91 for perovskites with tetragonal, rhombohedral, and
orthorhombic distortions respectively (distortions with more available
polarization directions are more accommodating)[8].
1.2.1 Domain Walls

Figure 1-5: Formation of 180o Ferroelectric Domain Walls: Illustration of
how 180o domain wall formation in a tetragonally distorted perovskite
ferroelectric can relieve electrostatic energy
A domain wall is typically classified by the angular difference between

the spontaneous polarization directions of the two domains it separates.
Besides 1800 domain walls, tetragonally distorted perovskites can have 900


10

domain walls while rombohedrally distorted perovskites can have 71o and
109o domain walls. As demonstrated in figures 1-5 and 1-6, 180o domain
walls can relieve both electrostatic as well as elastic energies. Because non180o domain walls separate domains with different spontaneous strain tensors,
these walls are ferroelastic as well as ferroelectric. Due to their ferroelastic
nature, the motion of non-180o domain walls contributes to the piezoelectric
response of a ferroelectric (180o domain wall movement does not) [5].

Figure 1-6: Strain Energy Relief by 90o Domain Wall Motion: Illustration of
how 90o domain wall motion in a tetragonally distorted perovskite
ferroelectric can relieve strain energy
1.2.2 Field-Induced Strain Mechanism and Domain Configurations in
Ferroelectric Ceramics
For ferroelectric ceramics, the electric field induced strain includes a
piezoelectric strain, an electrostrictive strain and a strain associated with a
ferroelectric domain reorientation. Both piezoelectric and electrostrictive
strains result from ionic shifts from their equilibrium positions by an electric
field application.