Piezoelectrics IPiezoelectrics I

Dr. Tanmoy MaitiDr. Tanmoy Maiti

MSE 689MSE 689

ModuleModule--PiezoPiezo--II

Classification of Piezoelectric

Materials

32 Point Groups

21 non centrosymmetric 11 centrosymmetric

20 piezoelectric (polarized under stress)

10 pyroelectric (spontaneous polarized)

Subgroup – ferroelectric (spontaneous, reversible polarized)

→→→→ polycrystalline ceramics

Origin of Piezoelectricity

• Piezoelectricity

– Discovered in 1880 by Jacque and Pierre Curie

– Materials exhibits electrical voltage with applied pressure

– Green “piezo” meaning “to press” – “pressure electricity”

– Converse piezoelectric effect (discovered shortly after)– Converse piezoelectric effect (discovered shortly after)

• Conversion of mechanical to electrical energy and vice versa

Mechanical

Electric

Direct Converse

Important PiezoImportant Piezo-- Effects and Effects and

CoefficientsCoefficients

PPV

Direct converse

iijkjk Edx =

Strain:

ijk

i

jk

jk

i dE

x

X

D==jkijki XdD =

Dielectric Displacement:

effectric piezoelectconverse

relation symmetrythe satifies ts coefficienric piezoelect

T

ijk

i

jk

ikjijk

E

ijk

jk

i

dE

x

dd

d

dX

D

=∂

∂

=

=∂

∂Direct Piezo coefficient

Piezoelectric CoefficientsPiezoelectric Coefficients

xjk

= Cijkl

Xkl

+ dijk

Ei

When X = 0

xjk

= dijk

Ei

Di= ε

ijE

j+ d

ijkX

jk

When E = 0,

Di= d

ijkX

jk

dijk

E= Direct piezoelectric coefficients =

Di

Xjk

=C /m

2

N /m2

=C

N

dijk

T = Converse piezoelectric effect =

xjk

Ei

=1

V /m=

m

V

So we need to prove that C

N=

m

V

Direct converse

N V

Proof :

Q = CV, Farad = C

V

F =q

1q

2

4πε0r

2=

C2

(F • m−1) • m

2=

C2

(C

V) • m

−1• m

2

=CV

m

N =CV

m

C

N=

m

V

Ferroelectrics

�� E.g. Barium Titanate or Lead Zirconate TitanateE.g. Barium Titanate or Lead Zirconate Titanate

�� Spontaneous Polarization is reversible by application of reverse Spontaneous Polarization is reversible by application of reverse

electric fieldelectric field

Curie Group

∞∞m

Symmetry is ∞∞m → no piezoelectricity

E Symmetry is ∞m → non-centricand permits piezoelectricity

Hysteresis curve

Curie Group

∞m

In the poling process, an intense dc field of the order of 20-25 kV/cm is applied to ferroelectric ceramic at a temperature sufficiently below its

curie point. The application of such an intense electric field on ferroelectric

Piezoelectricity in Ceramics:

intense electric field on ferroelectric ceramic changes its domain patterns which leads to a preferential alignment of the polarity in the direction of the applied electric field.

*Poling

Ferroelectric

Ceramics:

Thus Quartz which is not a ferroelectric material, can not exhibit piezoelectricity in polycrystalline form where as PZT, which is a ferroelectric, can show piezoelectricity after appropriate poling

Role of Domain Wall Motion in Piezoelectrics (Ferroelectrics)

180° domain wall motion:

PS PS

c

a a

c

PS PS

c

a a

c

No dimensional change, no contribution to piezoelectricity

under E field

a a a a

90° domain wall motion:

PSc

a

c

a

c

a

PS

PSc

c

a

c

a

PS

under E field

dimensional change, significant contribution to piezoelectricity

c>a

MATHEMATICAL TREATMENT OF FERROELECTRICS

TENSOR REPRESENTATION OF PHYSICAL PROPERTIES

x = d E

E and x are first-rank and second-rank tensors, respectively, the d should have a

third-rank tensor form represented as

xjk = ΣΣΣΣ dijk Ei (2.5)

ii

The d tensor is composed of three layers of the symmetrical matrices.

d111 d112 d113

1st layer (i = 1) d121 d122 d123

d131 d132 d133

d211 d212 d213

2nd layer (i = 2) d221 d222 d223 (2.6)

d231 d232 d233

d311 d312 d313

3rd layer (i = 3) d321 d322 d323

d331 d332 d333

REDUCTION OF THE TENSOR (MATRIX NOTATION)

A general third-rank tensor has 33 = 27 independent components. Since dijk is

symmetrical in j and k some of the coefficients can be eliminated, leaving 18

independent dijk coefficients; this facilitates the use of matrix notation.

So far all the equations have been developed in full tensor notation. But when

calculating actual properties, it is advantageous to reduce the number of suffixes as

much as possible. This is done by defining new symbols, for instance, d21 = d211 and

d14 = 2d123: The second and third suffixes in the full tensor notation are replaced by a

single suffix 1 to 6 in matrix notation, as follows:

Tensor notation 11 22 33 23,32 31,13 12,21

__________________________________________________

Matrix notation 1 2 3 4 5 6

x1 d11 d21 d31 E 1

x2 d12 d22 d32 E 2

x3 = d13 d23 d33 E 3 (2.21)

x4 d14 d24 d34

x5 d15 d25 d35

x6 d16 d26 d36

d31 =d 32

d33

d15 =d24

Matrix Notation

Di = dikXk’

xk’ = dik’Ei

i=1,2,3 and k’=1,2,3,4,5,6

i=j

i11

ijk

j=k

ik’

i1

d31=d32

d33

d15=d24

6mm identical to poled ceramic with ∞m

i22 i2i33 i3

kj ≠

i12 i6

i13 i5

i23 i4

Important Orientation Important Orientation

RelationsRelations

6mm or ∞m

d31 = d32

d33

d15 = d24d15 = d24

d15

Shear mode

Problem:Suppose that a shear stress is applied to a square crystal and the crystal is deformed as illustrated in Fig.1. Calculate the induced strain x5 ( = 2x31).

Fig. 1 Shear stress and strain configuration.

Solution

Since x5 = 2x31 = tan θ = θ and 1° = π /180 rad., x5 = 0.017.____________________________________________________

Problem:Barium titanate shows a tetragonal crystal symmetry (point group 4mm) at room temperature. Therefore, its piezoelectric constant matrix is :

(a) Calculate the induced strain under an electric field applied along the crystal c axis.(b) Calculate the induced strain under an electric field applied along the crystal a axis.crystal a axis.

Solution

is transformed intox1 = x2 = d31E3

x3 = d33E3

x4 = d15E2

x5 = d15E1, E6 = 0

(a) When E3 is applied, elongation in the c direction (x3 = d33E3, d33 > 0) and contaction in the a and b directions (x1 = x2 = d31E3, d31 < 0) are induced.induced.

(b) When E1 is applied, shear strain x5 (= 2x31) = d15E1 is induced.

Figure (a) illustrates a case of d15 >0 and x5 >0.

Crystal Orientation Dependence of Piezoelectric d33 in Pb(Zr,Ti)O3

PZT 40/60 Tetragonal PZT 60/40 Rhombohedral

High Strain Piezoelectrics (1)

d[001] /d[111] = 2.6

Rhombohedral compositions with the

<001> orientation may provide the

max d and k !

Substrate Applying electric field along [001] direction

d Tensor Form for 4mm Symmetry

For a third-rank tensor such as the piezoelectric tensor, the transformation due

to a change in coordinate system is represented by

d'ijk = Σ ailajmakn dlmn (2.15)

When the crystal has a 4-fold axis along z-axis, for example, the

transformation matrix is given by0 1 0

-1 0 0

0 0 1

or a12 = 1, a21 = -1, a33 = 1

Considering the tensor symmetry with m and n such that d123 = d132 and d213 = d231 (each matrix of the ith layer of the d tensor is symmetrical):

d‘111 = Σ a1la1ma1n dlmn

= a12a12a12 d222

= (+1)(+1)(+1) d222 = d222 d111 = d222

d‘122 = Σ a1la2ma2n dlmn

= a12a21a21 d211

= (+1)(-1)(-1) d211 = d211 d122 = d211

d‘133 = a12a33a33 d233

= (+1)(+1)(+1) d233 = d233 d133 = d233

d‘123 = a12a21a33 d213

= (+1)(-1)(+1) d213 = - d213 d123 = - d213

= d132 = - d231d‘131 = a12a33a12 d232

= (+1)(+1)(+1) d232 = d232 d131 = d232= (+1)(+1)(+1) d232 = d232

= d113 = d223d‘112 = a12a12a21 d221

= (+1)(+1)(-1) d221 = − d221 d112 = − d221

= d121 = − d212

d‘211 = a21a12a12 d122

= (-1)(+1)(+1) d122 = - d122 d211 = - d122

d122 = d211 = - d122 = 0

d Tensor Form for 4mm Symmetry -- Cont.

We can obtain:

d111 = d222 = d112 = d121 = d211 = d221 = d212 = d122

= d331 = d313 = d133 = d332 = d323 = d233

= d312 = d321 = 0

Calculating for all 18 dijk’s, including d222, d233, d223, d231, d212, d311, d322, d333, d323, d331, d312,

= d312 = d321 = 0d333 ≠ 0d311 = d322

d113 = d131 = d223 = d232

d123 = d132 = -d213 = -d231 (2.16)

0 0 0 0 d15 0

0 0 0 d15 0 0

d31 d31 d33 0 0 0

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