1. Introduction to Ferroelectric Materials and Devices 426415

2. Objectives To have basic knowledge of ferroelectric material
To understand piezoelectric effects
To describe its application
To understand lead zirconate titanate (PZT) solid solution system

3. What is this material ?

4. Piezoelectric effects
(1)
(3)
(2) 4

5. Piezoelectric effects
Direct effect
D = dT + TE
Converse effect
S = sET + dE
D is dielectric displacement = polarization, T is the stress, E is the electric field, S = the strain, s = the material compliance (inverse of modulus of elasticity),  = dielectric constant, d = piezoelectric (charge) constant

6. Piezoelectric constants
Piezoelectric Charge Constant (d)
The polarization generated per unit of mechanical stress applied to a piezoelectric material
alternatively
The mechanical strain experienced by a piezoelectric material per unit of electric field applied
Electro-Mechanical Coupling Factor
(i) For an electrically stressed component
k2 = stored mechanical energy
total stored energy
(ii) For a mechanically stressed component
k2 = stored electrical energy

8. Polarization
5 Basic polarizations
Dipole moment

9. Polarization Five basic types of polarisation:
(a) electronic polarisation of an atom,
(b) atomic or ionic polarisation of an ionic crystal,
(c) dipolar or orientational polarisation of molecules with asymmetry structure (H2O),
(d) spontaneous polarisation of a perovskite crystal,
and (e) interface or space charge polarisation of a dielectric material.
(Left-hand-side pictures illustrate the materials without an external electrical field and the right-hand-side pictures with an external electrical field, E.)

10. Crystallographic point group
32 crystallographic point groups. The remark “ i ” represents centrosymmetric crystal which piezoelectric effect is not exhibited, both remark “ * ” and “ + ” represent noncentrosymmetric crystal where the remark “ * ” indicates that piezoelectric effect may be exhibited and the remark “ + ” indicates that pyroelectric and ferroelectric effects may be exhibited.

11. Symmetry elements There are 3 types of symmetry operations: Rotation
Reflection
Inversion
An example of 4-fold rotation symmetry

12. Symmetry elements 1-Fold Rotation Axis 2-fold Rotation Axis

13. Symmetry elements Mirror Symmetry Mirror symmetry No mirror symmetry

14. Symmetry elements Center of Symmetry

15. Symmetry elements Center of Symmetry
In this operation lines are drawn from all points on the object through a point in the center of the object, called a symmetry center (symbolized with the letter "i").
If an object has only a center of symmetry, we say that it has a 1 fold rotoinversion axis.  Such an axis has the symbol , as shown in the right hand diagram above

16. Symmetry elements Rotoinversion
Combinations of rotation with a center of symmetry perform the symmetry operation of rotoinversion.
2-fold Rotoinversion - The operation of 2-fold rotoinversion involves first rotating the object by 180o then inverting it through an inversion center. 
This operation is equivalent to having a mirror plane perpendicular to the 2-fold rotoinversion axis.  A 2-fold rotoinversion axis is symbolized as a 2 with a bar over the top, and would be pronounced as "bar 2". 
But, since this the equivalent of a mirror plane, m, the bar 2 is rarely used.

17. Symmetry elements Rotoinversion
3-fold Rotoinversion - This involves rotating the object by 120o (360/3 = 120), and inverting through a center.  A cube is good example of an object that possesses 3-fold rotoinversion axes.  A 3-fold rotoinversion axis is denoted as (pronounced "bar 3").  Note that there are actually four axes in a cube, one running through each of the corners of the cube. If one holds one of the axes vertical, then note that there are 3 faces on top, and 3 identical faces upside down on the bottom that are offset from the top faces by 120o. 

18. Symmetry elements Combinations of Symmetry Operations
As should be evident by now, in three dimensional objects, such as crystals, symmetry elements may be present in several different combinations.  In fact, in crystals there are 32 possible combinations of symmetry elements.  These 32 combinations define the 32 Crystal Classes.  Every crystal must belong to one of these 32 crystal classes. 

19. Crystal system
(a) A simple square lattice. The unit cell is a square with a side a.
(b) Basis has two atoms.
(c) Crystal = Lattice + Basis. The unit cell is a simple square with two atoms.
(d) Placement of basis atoms in the crystal unit cell.

20. Crystal system
(a) A simple square lattice. The unit cell is a square with a side a.
(b) Basis has two atoms.
(c) Crystal = Lattice + Basis. The unit cell is a simple square with two atoms.
(d) Placement of basis atoms in the crystal unit cell.

21. 7 crystal systems and 14 bravais lattices
The seven crystal systems (unit cell geometries) and fourteen Bravais lattices.

22. Crystal system and symmetry elements
axis lengths
angles between axes
common symmetry
elements
triclinic
a ≠ b ≠ c
 ≠  ≠  ≠ 90°
1-fold rotation w/ or
w/out i
monoclinic
 =  = 90°, β >90°
2-fold rotation and/or 1m
orthorhombic
 =  =  = 90°
3 2-fold rotation axes
and/or 3 m
hexagonal
a1 = a2 = a3, a ≠ c
60° btw a’s,  = 90o
1 3-fold or 6-fold axis
tetragonal
a = b ≠ c
 =  = = 90°
1 4-fold rotation or
rotoinversion axis
cubic
4 3-fold axes

23. Crystal system and symmetry elements
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

24. Crystal directions
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

25. Crystal planes
Labeling of crystal planes and typical examples in the cubic lattice
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

26. Crystal planes
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

27. Zr+4 or Ti+4
Pb+2
O-2

28. Polarization
Classification of piezoelectric, pyroelectric and ferroelectric effects based on the symmetry system.

29. Crystal with a centre of symmetry
A NaCl-type cubic unit cell has a center of symmetry.
In the absence of an applied force, the centers of mass for positive and negative
ions coincide.
(b) This situation does not change when the crystal is strained by an applied force.

30. Noncentrosymmetric crystal
A hexagonal unit cell has no center of symmetry. (a) In the absence of an applied force the centers of mass for positive and negative ions coincide. (b) Under an applied force along y the centers of mass for positive and negative ions are shifted which results in a net dipole moment P along y. (c) When the force is along a different direction, along x, there may not be a resulting net dipole moment in that direction though there may be a net P along a different direction (y).

31. Lead Zirconate Titanate (Pb(Zrx,Ti1-x)O3 or PZT) System
Cubic
Pb+2
O-2
Zr+4 or Ti+4
Perovskite
Rhombohedral High Temperature P
Tetragonal
Rhombohedral Low Temperature
MPB
PZT Solid Solution Phase Diagram
Zr/Ti ratio 52/48 MPB (Morphotropic Phase Boundary)

32. Lead Zirconate Titanate (Pb(Zrx,Ti1-x)O3 or PZT) System
PZT (xPbZrO3 – (1-x)PbTiO3) 
 Binary Solid Solution 
PbZrO3 (antiferroelectric matrial with orthorhombic structure)
and
PbTiO3 (ferroelectric material with tetragonal perovskite structure)
Perovskite Structure (ABO3) with Ti4+ and Zr4+ ions “randomly” occupying the B-sites
Important Transducer Material (Replacing BaTiO3)
Higher electromechanical coupling coefficient (K) than BaTiO3
Higher Tc results in higher operating and fabricating temperatures
Easily poled
Wider range of dielectric constants
relatively easy to sinter at lower temperature than BaTiO3
form solid-solution compositions with several additives which results in a wide range of tailored properties

33. Lead Zirconate Titanate (Pb(Zrx,Ti1-x)O3 or PZT) System
Composition dependence of dielectric constant (K) and electromechanical planar coupling coefficient (kp) in PZT system 
This shows enhanced dielectric and electromechanical properties at the MPB
Increased interest in PZT materials with MPB-compositions for applications

34. Lead Zirconate Titanate (Pb(Zrx,Ti1-x)O3 or PZT) System
Advantages of PZT Solid-Solution System
High Tc across the diagram leads to more stable ferroelectric states over wide temperature ranges
There is a two-phase region near the Morphotropic Phase Boundary (MPB) (52/48 Zr/Ti composition) separating rhombohedral (with 8 domain states) and tetragonal (with 6 domain states) phases
In the two-phase region, the poling may draw upon 14 orientation states leading to exceptional polability
Near vertical MPB results in property enhancement over wider temperature range for chosen compositions near the MPB

35. Compositions and Modifications of PZT System
1. Effects of composition and grain size on properties
MPB compositions
(Zr/Ti = 52/48) 
Maximum dielectric and piezoelectric properties
Selection of Zr/Ti can be used to tailor specific properties
High kp and er are desired
 Near MPB compositions OR
High Qm and low er are desired
 Compositions away from MPB
Grain Size
(composition and processing) 
Fine-Grain ~ 1 mm or less
Coase-Grain ~ 6-7 mm
Some oxides are grain growth inhibitor (i.e. Fe2O3)
Some oxides are grain growth promoter (i.e. CeO2)
Dielectric and piezoelectric properties are grain-size dependent

36. Compositions and Modifications of PZT System
2. Influences of low level “off-valent” additives (0-5 mol%) on dielectric and piezoelectric properties
Two main groups of additives:
1. electron acceptors (charge on the replacing cation is smaller)
(A-Site:K+, Rb+ ; B-Site: Co3+, Fe3+, Sc3+, Ga3+, Cr3+, Mn3+, Mn2+, Mg2+, Cu2+)
(Oxygen Vacancies)
Reduce both dielectric and piezoelectric responses
Increase highly asymmetric hysteresis and larger coercivity
Much larger mechanical Q
“Hard PZT”
2. electron donors (charge on the replacing cation is larger)
(A-Site: La3+, Bi3+, Nd3+; B-Site: Nb5+, Ta5+, Sb5+)
(A-Site Vacancies)
Enhance both dielectric and piezoelectric responses at room temp
Under high field, symmetric unbiased square hysteresis loops
low electrical coercivity
“Soft PZT”

37. Conclusion
Piezoelectric effects 1. mechanical energy  electrical energy :sensors
2. electrical energy  mechanical energy :actuators
3. noncentrosymmetric crystal, perovskite structure
Lead Zirconate Titanate Pb(Zrx,Ti1-x)O3  PZT
near MPB  high piezoelectric response (high K and d)
Hard PZT  additive = electron acceptors (A-Site:K+, Rb+ ; B-Site: Co3+, Fe3+, Sc3+, Ga3+, Cr3+, Mn3+, Mn2+, Mg2+, Cu2+)
low piezoelectric response
Soft PZT  additive = electron donors (A-Site: La3+, Bi3+, Nd3+; B-Site: Nb5+, Ta5+, Sb5+)
high piezoelectric response

38. Modified PZT System “Hard PZT” Materials “Soft PZT” Materials
Curie temperature above 300 C
NOT easily poled or depoled except at high temperature
Small piezoelectric d constants
Good linearity and low hysteresis
High mechanical Q values
Withstand high loads and voltages
“Soft PZT” Materials
Lower Curie temperature
Readily poled or depoled at room temperature with high field
Large piezoelectric d constants
Poor linearity and highly hysteretic
Large dielectric constants and dissipation factors
Limited uses at high field and high frequency

39. Domain structure Zr+4 or Ti+4 Pb+2 O-2 Tc ~ 350 oC
(a) Domain structure of a tetragonal ferroelectric ceramic (lead zirconate titanate) with 180o and 90o domain walls is revealed by etching in HF and HCl solution. The formation of parallel lines in the grains of the ceramic (a) is due to 90o orientation of the polar direction. (b) A schematic drawing of 90o and 180o domains in a ferroelectric ceramic.

40. ferroelectric domain switching
The application of an electric field causes (a) the reorientation of a spontaneous polarisation, Ps, in a unit cell to the field direction; (b) the sum of the microscopic piezoelectricity of the unit cells result in the macroscopic piezoelectricity of piezoceramics.

41. Compositions and Modifications of PZT System
2. Modification by element substitution
Element substitution  cations in perovskite lattice (Pb2+, Ti4+, and Zr4+) are replaced partially by other cations with the same chemical valence and similar ionic radii and solid solution is formed
Pb2+ substituted by alkali-earth metals, Mg2+, Ca2+, Sr2+, and Ba2+  
PZT replaced partially by Ca2+or Sr2+
Tc  BUT kp, e33 , and d31  
Shift of MPB towards the Zr-rich side 
Density  due to fluxing effect of Ca or Sr ions 
Ti4+ and Zr4+ substituted by Sn4+ and Hf4+ , respectively 
Ti4+ replaced partially by Sn4+
c/a ratio decreases with increasing Sn4+ content 
 Tc  and  stability of kp and e33 

42. (3) Polarization Electric Field Reference:
3. Ferroelectric Materials, n.d. DoITPoMs teaching and learning package, viewed 8 December 2008, <
Electrode
Electrode
Polarization
Electrode
Electric Field
Electrode
Electrode 42

43. Polarization Electric Field 43 Electrode Electrode Electrode Electrode

44. Dielectric Hysteresis loop
A 2 dimensional schematic sketch of an ideal polarisation hysteresis loop. The dashed line presents the initial polarisation process of the thermally unpoled ceramic. The domain orientation state is represented by arrows in the boxes.

45. Butterfly Hysteresis loop
A 2 dimensional schematic sketch of an ideal butterfly hysteresis loop. The dashed line presents the initial polarisation process of the thermally unpoled ceramic. The domain orientation state is represented by arrows in the boxes

46. ferroelastic domain switching
(a) 90o switching of a polarisation in a unit cell induced by a mechanical loading; the polarisation direction of each unit cell is represented by a black arrow beside it. (b) 90o domain switching under a compressive loading in a polycrystal; the polarisation direction is represented by the black arrow in each grain.

47. ferroelastic domain switching
represents unsysmetric ferroelastic hysteresis and domain switching states after applying tensile and compressive loading. The sketch simply simulates the possible position of c-axis orientation with in a unit cell.

48. ferroelastic domain switching? ?
Compressive stress-strain curve of poled soft and hard PZT

49. Converse Piezoelectric Effect
Di =Pi= dij Tj
Pi = induced polarization along the i direction, dij = piezoelectric coefficients, Tj = mechanical stress along the j direction
Converse Piezoelectric Effect
Sj = dij Ei
Sj = induced stain along the j direction, dij = piezoelectric coefficients, Ei = electric field along the i direction

50. Electro-Mechanical Coupling Factor
k = electromechanical coupling factor
Electro-Mechanical Coupling Factor
k = electromechanical coupling factor

51. Piezoelectric Equations and Constants
Piezoelectric Charge Constant (d)
The polarization generated per unit of mechanical stress applied to a piezoelectric material
alternatively
The mechanical strain experienced by a piezoelectric material per unit of electric field applied
Piezoelectric Voltage Constant (g)
The electric field generated by a piezoelectric material per unit of mechanical stress applied
The mechanical strain experienced by a piezoelectric material per unit of electric displacement applied.

52. Piezoelectric Materials

53. Piezoelectric Figures of merit*
*A figure of merit is a quantity used to characterize the performance of a device

54. Piezoelectric Figures of merit

55. Piezoelectric Figures of merit
Coupling factor K

56. Piezoelectric Figures of merit

57. Piezoelectric Figures of merit
The mechanical quality factor , QM
= (Strain in phase with stress)/(Strain out of phase with stress)
High QM  low energy lost to mechanical damping. So piezoelectric material with high QM is desirable in a piezoelectric driver or resonator

58. Piezoelectric constants
Permittivity 

59. Piezoelectric constants

60. Basic Piezoelectric mode
The piezoelectric constants of a ferroelectric material poled in 3-direction. (a) shows d33 and d31-effect and (b) shows d15-effect.

61. Basic Piezoelectric mode

62. Piezoelectric transducers are widely used to generate ultrasonic waves in solids and also to detect such mechanical waves. The transducer on the left is excited from an ac source and vibrates mechanically. These vibrations are coupled to the solid and generate elastic waves. When the waves reach the other end they mechanically vibrate the transducer on the right which converts the vibrations to an electrical signal.

63. Piezoelectric Voltage Coefficient
E = gT
E = electric field, g = piezoelectric voltage coefficient T = applied stress
g = d/(or)

64. Piezoelectric Spark Generator
The piezoelectric spark generator, as used in various applications such as lighters and car ignitions, operates by stressing a piezoelectric crystal to generate a high voltage which is discharged through a spark gap in air as schematically shown in picture (a). Consider a piezoelectric sample in the form of a cylinder as in this picture. Suppose that the piezoelectric coefficient d = 250 x mV-1 and r = The piezoelectric cylinder has a length of 10 mm and a diameter of 3 mm. The spark gap is in air and has a breakdown voltage of about 3.5 kV. What is the force required to spark the gap? Is this a realistic force?
The piezoelectric spark generator.

65. Piezoelectric Quartz Oscillators
When a suitably cut quartz crystal with
electrodes is excited by an ac voltage as
(a), it behaves as if it has the equivalent
Circuit in (b).
(c) and (d) The magnitude of the
impedances Z and reactance (both
between A and B) versus frequency,
neglecting losses.

66. Mechanical Resonant Frequency
fs = mechanical resonant frequency, L = mass of the transducer, C = stiffness
Antiresonant Frequency
fa = antiresonant frequency, L = mass of the transducer, C is Co and C in parallel, where Co is the normal parallel plate capacitance between electrodes
For oscillators, the circuit is designed so that oscillations can take place only when the crystal in the circuit is operated at fs

67. Design of Buzzer

68. Design of Buzzer

69. A typical 1 MHz quartz crystal has the following properties:
fs = 1 MHz, fa = MHz, Co = 5 pF, R = 20 .
What is C in the equivalent circuit of the crystal? What is the quality factor Q of the crystal, given that

70. Piezoelectric measurement