Download The Figure shows that the open-circuit voltage V (and hence the

The Figure shows that the open-circuit
voltage V (and hence the field strength
E) is proportional to compressive stress
T up to a maximum of 25 kV, which
occurs at a stress of 50 MPa.
Induced electrical field
E=
g=
E
σ
25000V
= 1.25 × 106V .m −1
0.02m
=
induced _ electric _ field
mechanical _ stress _ applied
1.25 × 106V .m −1
V
g= =
=
0
.
025
50 × 106 Pa
σ
Pa × m
E
Definitions of frequent use
The piezoelectric charge
P
Induced _ Polarization
d= =
constant, (d), polarization
σ Mechanical _ Stress _ applied
generated per unit of mechanical
ε mechanical _ strain _ obtained
stress (direct), or, mechanical
d= =
strain experienced by a
E
electric _ field _ applied
piezoelectric per unit of electric
field applied (converse).
E
induced _ electric _ field
The piezoelectric voltage
g= =
σ mechanical _ stress _ applied
constant, (g), is the electric
field generated by a
mechanical _ strain − obtained
ε
piezoelectric material per unit of g = =
D electric _ displacement _ applied
mechanical stress applied or,
alternatively, is the mechanical
strain experienced by a
d
g=
piezoelectric material per unit of
ε rεO
electric displacement applied.
Example (Piezoelectric)
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 discharge through a spark gap in air (see
F
(a)
figure below).
The breakdown field for
air is 3x106V.m-1. If you
consider a gap of 1mm
it is about 3000V.
F
A
L
Piezoelectric
Piezoelectric
V
Piezoelectric
F
F
(b)
Fig. 7.39: The piezoelectric spark generator
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Additional Notes
The ratio of strain to electric field is called the “d” constant for a piezoelectric.
The ratio of electric field generated to stress applied is called the “g” constant
(piezoelectric voltage coefficient) for a piezoelectric, where E(V/m) is the
electric field, ε is the strain , σ is the stress (Pa)
The “g” and “d” are related to the dielectric constant as follows: where εr is
the dielectric constant and εO is the permittivity under vacuum (8.85x10-12F/m).
ε
strain
d= =
E electric _ field _ generated
E electric _ field _ generated
g= =
σ
stress
g=
d
ε rεO
Consider a piezoelectric sample in the form of a cylinder (see figure).
Suppose that the piezoelectric coefficient d=250x10-12m.V-1 and εr=1000.
The piezoelectric has a length of 10mm and a diameter of 3mm. The
spark gap is in air and has a breakdown voltage of about 3.5kV. What is
the force required to spark the gap? Is this a realistic force?
Solution
250 × 10 −12 m.V −1
=
g=
ε r ε O (1000 )× 8.85 × 10 −12 F .m −1
d
(
g = 2.8 × 10 − 2 m 2 .(V .F ) −1
V 3500V
= 3.5 × 105V .m −1
E= =
L 0.01m
F
)
A
L
Piezoelectric
F
V
E
3.5 × 105V .m −1
g = →σ = =
σ
g 2.8 × 10 − 2 (V .F ) −1 .m 2
E
σ = 1.25 ×107 V 2 .F .m −3 = 1.25 ×107 N .m − 2
Farad = F units (kg-1 m-2 s4 A2)
Volts=V units (kg m2 s-3 A-1)
Newton=N units (kg m s-2)
Force F
π
7
= → F = σ × A = 1.25 × 10 × × 0.0032
σ=
Area A
4
F = 88.35N
This force (about 9kg-f) can be applied by squeezing by hand an appropriate
lever arrangement.
The force must be applied quickly because the piezoelectric charge
generated will leak away (become neutralized).
The voltage generated can be increased (or the force needed reduced) by
using two piezoelectric crystals back to back.
Piezoelectric Constants
Because a piezoelectric is anisotropic, its physical constants (elasticity,
permittivity, etc.) are tensor quantities and relate to both the direction of the
applied stress or electric field and the directions perpendicular to these.
Each constant generally has two subscripts that indicate the directions of
the two related quantities, such as stress and strain for elasticity.
Piezoelectric Ceramics:
The direction of positive polarization is
made to coincide with the Z-axis of a
orthogonal system of X, Y, and Z axes.
Direction X, Y, or Z is represented by the
subscript 1, 2, or 3, respectively, and
shear about one of these axes is
represented by the subscript 4, 5, or 6,
respectively.
P
Induced _ Polarization
Piezoelectric Charge d = σ = Mechanical _ Stress _ applied
Constant (d):
d ab
Example:
ε
mechanical _ strain _ obtained
d= =
E
electric _ field _ applied
a indicates the direction of induced polarization of the material when
the electric field is zero or the direction of the applied electric field
b indicates the direction of the applied stress or the direction of the
strain.
d13 induced polarization voltage measured in the direction 1 (perpendicular
to the poling direction) when a stress is applied in a direction 3 or
or induced strain in the direction 3 when an electric field is applied in the
direction 1 (perpendicular to the poling direction).
d ab
d33 induced polarization in direction 3 (parallel to direction in which
ceramic element is polarized) per unit stress applied in direction 3
or induced strain in direction 3 per unit electric field applied in direction 3
d15 induced polarization in direction 1 (perpendicular to direction in which
ceramic element is polarized) per unit shear stress applied about direction
2 (direction 2 perpendicular to direction in which ceramic element is
polarized)
or induced shear strain about direction 2 per unit electric field applied in
direction 1
Piezoelectric Voltage
Constant (g)
g ab
E
induced _ electric _ field
g= =
σ mechanical _ stress _ applied
ε
mechanical _ strain − obtained
g= =
D electric _ displacement _ applied
a indicates the direction of the induced electric field on the
material or the direction of the applied electric displacement.
b indicates the direction of the applied stress or the direction of
the obtained strain.
Example:
g31 induced electric field in direction 3 (parallel to direction in which ceramic
element is polarized) per unit stress applied in direction 1 (perpendicular to
direction in which ceramic element is polarized)
or induced strain in direction 1 per unit electric displacement applied in
direction 3.
g ab
g33 induced electric field in direction 3 (parallel to direction in which ceramic
element is polarized) per unit stress applied in direction 3
or induced strain in direction 3 per unit electric displacement applied in
direction 3
g15 induced electric field in direction 1 (perpendicular to direction in which
ceramic element is polarized) per unit shear stress applied about direction 2
(direction 2 perpendicular to direction in which ceramic element is polarized)
or induced shear strain about direction 2 per unit electric displacement
applied in direction 1
If Tj is the applied mechanical stress along some j direction and Pi is the
induced polarization along some i direction, we relate them via
Pi = dij Tj
where dij are called piezoelectric coefficients and T can represent either
tensile or shear stresses.
An equivalent relation between the strain Sj along the j direction and the
electric field Ei along the i direction is given by
Sj = dij Ei
Effect of Stress on Tetrahedral groups in Piezoelectrics
(e.g. Quartz)
Piezoelectric crystals are essentially electromechanical transducers as
they convert electrical signals to mechanical signals, strain or vice versa.
Typical engineering applications: ultrasonic transducers, microphones,
sonar detectors, accelerometers, frequency control of oscillators and
filters, monitoring of thin film deposition.
Eg. In phonographic pick-ups; stylus traverses grooves of record
pressure variation imposed on a piezoelectric material located in
cartridge transformed into electrical signal amplified and broadcasted
through speaker.
Efficiency of conversion between electrical and mechanical energy is
given by the electromechanical conversion factor K defined in terms of
K2 by
K2 =
mechaniucal _(electrical ) _ energy _ output
electrical _(mechanical ) _ energy _ input
Characteristics:
Light weight and compact.
Inexpensive
Relative linear field-strain relations at low drive levels.
Broadband drive capabilities
Very high set-point accuracy
Actuator and sensor capabilities
Due to the non-centrosymmetric nature of ferroelectric materials they exhibit
hysteresis and constitutive non-linearities at all drive levels. For low drive
regimes these effects can be mitigated through feedback mechanisms. For
high drive regimes, it is necessary to employ charge or current control.
Electrostatic transducers constructed from relaxors ferroelectric materials are
advantageous due to the fact that they exhibit minimal hysteresis.
Unlike piezoelectric materials, electrostrictive compounds are not poled and
hence exhibit few aging effects.
Direct piezoelectric effect:
In this effect an electric polarization “P” arises
as a result of an applied stress σ.
The applied stress is in reality a stress tensor
given by
Then, the polarization in the i direction (axis),
is related to the stress σjk by the coefficient
dijk .
For example:
⎡σ xx τ yx τ zx ⎤
⎥
⎢
σ = ⎢τ xy σ yy τ zy ⎥
⎢τ xz τ yz σ zz ⎥
⎦
⎣
D (C/m2) ∝ Εο (electric field V/m)
D = εΕο+P D=dielectric displacement
Pi = ∑ d ijk × σ jk
jk
P1 = d111σ 11 + d112σ 12 + d113σ 13 + d121σ 21 + d122σ 22 + d123σ 23 + d131σ 31 + d132σ 32 + d133σ 33
P2 = d 211σ 11 + d 212σ 12 + d 213σ 13 + d 221σ 21 + d 222σ 22 + d 223σ 23 + d 231σ 31 + d 232σ 32 + d 233σ 33
P3 = d 311σ 11 + d 312σ 12 + d 313σ 13 + d 321σ 21 + d 322σ 22 + d 323σ 23 + d 331σ 31 + d 332σ 32 + d 333σ 33
The 27 dijk are the piezoelectric moduli and form a third rank tensor.
As dijk = dikj, there are up to 18 independent moduli.
⎛ P1 ⎞ ⎡ d111
⎜ ⎟ ⎢
⎜ P2 ⎟ = ⎢d 211
⎜ P ⎟ ⎢d
⎝ 3 ⎠ ⎣ 311
d122
d 222
d 322
d133
d 233
d 333
2d123
2d 223
2d 323
2d113
2d 213
2d 313
⎛ σ 11 ⎞
⎟
⎜
⎜ σ 22 ⎟
2d121 ⎤⎜
⎟
σ
2d 221 ⎥⎥⎜ 33 ⎟
⎜ σ 23 ⎟
2d 321 ⎥⎦⎜
⎟
σ
⎜ 13 ⎟
⎜σ ⎟
⎝ 21 ⎠
P1 = d111σ 11 + d112σ 12 + d113σ 13 + d121σ 21 + d122σ 22 + d123σ 23 + d131σ 31 + d132σ 32 + d133σ 33
Converse piezoelectric effect:
In this effect a strain arises as a result of an applied
electric field. The moduli are the same as for the direct
effect.
ε jk = ∑ dijk × Ei
i
ε11 = d111E1 + d 211E2 + d 311E3
ε 22 = d122 E1 + d 222 E2 + d 322 E3
ε 33 = d133 E1 + d 233 E2 + d 333 E3
ε12 = d112 E1 + d 212 E2 + d 312 E3
ε13 = d113 E1 + d 213 E2 + d 313 E3
ε 23 = d123 E1 + d 223 E2 + d 323 E3
ε 21 = d121E1 + d 221E2 + d 321E3
ε 31 = d131E1 + d 231E2 + d 331E3
ε 32 = d132 E1 + d 232 E2 + d 332 E3
In a matrix form
⎛ ε11 ⎞ ⎡ d111
⎜ ⎟ ⎢
⎜ ε 22 ⎟ ⎢d122
⎜ ε ⎟ ⎢d
⎜ 33 ⎟ = ⎢ 133
⎜ ε 23 ⎟ ⎢ d123
⎜ ⎟ ⎢d
⎜ ε13 ⎟ ⎢ 113
⎜ ε ⎟ ⎢d
⎝ 12 ⎠ ⎣ 112
d 211
d 222
d 233
d 223
d 213
d 212
d 311 ⎤
⎥
d 322 ⎥
⎛ E1 ⎞
d 333 ⎥⎜ ⎟
⎥ ⎜ E2 ⎟
d 323 ⎥⎜ ⎟
E3 ⎠
⎝
d 313 ⎥
⎥
d 312 ⎥⎦
Crystal Orientation
The direction in which tension or compression develops polarization parallel to
the strain is called the piezoelectric axis. In quartz, this axis is knows as the "Xaxis", and in poled ceramic materials such as PZT the piezoelectric axis is
referred to as the "Z-axis". From different combinations of the direction of the
applied field and orientation of the crystal it is possible to produce various
stresses and strains in the crystal.
For example, an electric field applied perpendicular to the piezoelectric axis will
produce elongation along the axis as shown. If, however, the electric field is
applied parallel to the piezoelectric axis, a shear motion is induced.
Neumann’s Principle
This is the most important concept in crystal physics. It states; ……………...
the symmetry of any physical property of a crystal must include the
symmetry elements of the point group of the crystal. This means that
measurements made in symmetry-related directions will give the same property
coefficients.
Example: NaCl belongs to the m3m group . The [100] and [010] directions are
equivalent.
Since these directions are physically the
same, it should be expected that
measurements of permittivity, elasticity or
any other physical property will be the same
in these two directions.
Piezoelectric Symmetry Groups
In 20 out of the 21 acentric point groups (only the 432 is excluded) the
application of a stress (σhk ) along a suitable direction generates an electric
dipole (P).
The relationship between the dipole Pi and the stress (σhk) is
Pi = ∑ d ihkσ hk
h, k
The dihk are the components of a third rank
tensor known as piezoelectric tensor.
27 components are expected for a third rank tensor dihk.
However, the stress tensor is symmetrical so dihk=dikh and the number
of components reduces to 18 (triclinic crystals).
Crystal symmetry constrain the 18 components in triclinic crystals to 10 or
8 in monoclinic crystals, 9 in orthorhombic, 7-6 in tetragonal, 6-4 or 2 in
trigonal, 4-2 or 1 in hexagonal and 1 in cubic.
Quartz (Point Group 32)
⎡d11 − d11 0 d14
⎢0
0
0 0
⎢
⎢⎣ 0
0
0 0
0
− d14
0
0 ⎤
− 2d11 ⎥⎥
0 ⎥⎦
Applications of ferroelectrics
The world market for ferroelectric materials and devices is in the range
$20-30billion per annum.
Capacitors
The widest application of ferroelectrics (not making use of ferroelectricity
but simply the high dielectric constant) is in capacitors. BaTiO3 has
cornered more than 50% of the ceramic capacitor market. For a given
volume, a BaTiO3 capacitor has 100-1000 times the capacitance of a
linear dielectric.
Ferroelectric memories (FeRAM)
Ferroelectrics exhibit bi-stable polarization (±Pr when Ε = 0). Hence,
binary memory systems can be designed based on the different
polarization states, +Pr =0, -Pr = 1.
e.g. computer RAM.
In FeRAM data (in the form of polarization state +Pr = 0, -Pr = 1) is
“recorded” on the material and is non-volatile (NV-RAM) – it is retained
when the voltage is turned off.
Square hysteresis loops are required for memory devices since it is
easier to distinguish between the 2 remanent polarization states of the
material ± Pr.
Hysteresis loops have different shapes. Psat ≠ Ps for a strongly sigmoidal
loop, whereas Psat = Ps = Pr for a “square” loop.
The applied field (i.e. read / write operations) is controlled by field effect
transistors (millions on a chip). They address the ferroelectric cells and
isolate them from their neighbors.
Switching occurs by domain wall motion – must be fast to compete with
Si-based memories where switching occurs by electron motion.
The materials requirements for FeRAM are high κ, high Psat (and Pr),
small and high Tc (operable at RT and slightly above). Examples are
LiNbO3, PbTiO3, Pb(ZrxTi1-x)O3 (PZT) SrBi2Ta2O9 (SBT). The materials
need to be made in thin film form (few hundred nm) in order to minimize
switching voltage.
The wide scale application of FeRAM devices will depend on the
successful materials science, namely optimization of composition,
microstructure, interface quality with the electrodes, and controlling the
nano-scale domain wall pinning defects. Target applications of FeRAM
include cell phones, smart cards, video games.