UCSD NANO106 - 09 - Piezoelectricity and Elasticity

Piezoelectricity and ElasticityShyue Ping OngDepartment of NanoEngineeringUniversity of California, San Diego

Piezoelectricity¡ Piezoelectricity refers to the linear coupling between mechanical stress and

electric polarization (the direct piezoelectric effect) or between mechanical strain and applied electric field (the converse piezoelectric effect). Equivalence between the direct and converse effects can be shown using Maxwell’s relations:

¡ Principal piezoelectric coefficient, d, relates polarization P to stress σ in the direct effect and strain ε to electric field E.

¡ Units of d: [C/N] or [m/V]

¡ Typical values for useful piezoelectric materials: ~ 1 pC/N for quartz to ~ 1000 pC/N for PZT (lead zirconate titanate).

NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 9

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dijk = −∂G2

∂Ei∂σ jk

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'((T

=∂Pi∂σ jk

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'((E,T

=∂ε jk∂Ei

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'(X,T

Direct piezoelectric effect¡ The piezoelectric moduli is a rank-3 tensor:

¡ It therefore transforms as

¡ As previously shown, the piezoelectric coefficients are zero for all centrosymmetric crystals (inversion center).


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Pi = dijkσ jk

dijk!i = cilcjmckndlmn

Voigt notation for piezoelectric moduli

¡ The piezoelectric moduli always act in pairs.

¡ Earlier, we have shown that 3x3 symmetric tensors such as stress and strain can be represented as a 6-element vector. Using the same concept, we can simplify the piezoelectric moduli to a 3x6 matrix as follows.

¡ However, here we encounter a problem that we cannot apply a without a . In order to go to a matrix notation, the piezoelectric moduli is defined as:


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Pi = dijkσ jk = di11σ11 + di22σ 22 + di33σ 33 + di23σ 23 + di32σ 32…(total of 27 elements!)

Pi = dijσ j = di1σ1 + di2σ 2 + di3σ 3 + di4σ 4 + di5σ 5 + di6σ 6

σ 23

σ 32

dim =dijk for j=k

dijk + dikj for j≠ k

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Factor of 2 is absorbed into the matrix, which has 18 elements instead of 27.

Voigt notation for piezoelectric moduli , contd.


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P1P2P3

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&&&&=

d11 d12 d13 d14 d15 d16d21 d22 d23 d24 d25 d26d31 d32 d33 d34 d35 d36

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&&&&

σ1σ 2

σ 3

σ 4

σ 5

σ 6

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&&&&&&&&

Converse piezoelectric moduli¡ The same coefficients apply to the converse piezoelectric effect (note

order of indices!)

¡ Similarly,

¡ Again, we note that by definition,


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ε jk = dijkEi

ε11 = d111E1 + d211E2 + d311E3ε22 = d122E1 + d222E2 + d322E3ε33 = d133E1 + d233E2 + d333E3ε23 = d123E1 + d223E2 + d323E3ε32 = d132E1 + d232E2 + d332E3

ε23 = ε32

ε23 +ε32 = (d123 + d132 )E1 + (d223 + d232 )E2 + (d323 + d332 )E3 = γ4

Voigt notation for converse piezoelectric effect¡ Recall that we earlier defined the vector notation for the strain tensor

to be the engineering shear strains (2 x strain) for the off-diagonals elements. This in fact allows us to retain the same form of the matrix for the converse piezoelectric effect as the direct piezoelectric effect!


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ε1ε2ε3γ4γ5γ6

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&&&&&&&&

=

d11 d21 d31d12 d22 d32d13 d23 d33d14 d24 d34d15 d25 d35d16 d26 d36

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&&&&&&&&

E1E2E3

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&&&&

Symmetry restrictions on piezoelectric moduli¡ As there are many more terms, we are not going to go

through the exercise of deriving all the possible piezoelectric moduli tensors / matrices.

¡ We will just go through a few examples to show how these can be derived, and state the different forms.

¡ Note that while the Voigt form is useful for performing computations, the information about how the tensor transforms is “lost”. You will need to go back to the full tensor form to determine the transformation.


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d13 in monoclinic crystals¡ Consider monoclinic crystals with a two-fold rotation axis (2):

¡ For monoclinic crystals with a mirror m about the a-c plane:


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Blackboard

Blackboard

Examples of piezoelectric moduli matrices


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Triclinic (P1) Orthorhombic 222

Orthorhombic 2mm

Trigonal 32

Cubic 432

d11 d12 d13 d14 d15 d16d21 d22 d23 d24 d25 d26d31 d32 d33 d34 d35 d36

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

d11 −d11 0 d14 0 00 0 0 0 −d14 −2d110 0 0 0 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

0 0 0 d14 0 00 0 0 0 d25 00 0 0 0 0 d36

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

0 0 0 0 d15 00 0 0 d24 0 0d31 d32 d33 0 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Monoclinic


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0 0 0 d14 0 d16d21 d22 d23 0 d25 00 0 0 d34 0 d36

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Point group 2

d11 d12 d13 0 d15 00 0 0 d24 0 d26d31 d32 d33 0 d35 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Point group m

Examples of Piezoelectric Materials¡ Quartz

¡ Consider the converse piezoelectric effect when a field of 100 V/cm is applied in the X1 direction

¡ This is an extremely small strain. But quartz has the advantage of being low cost, chemically stable and mechanically stable.


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2.3 −2.3 0 −0.67 0 00 0 0 0 0.67 −4.60 0 0 0 0 0

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$$$

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'''×10−12C / N

Piezoelectric moduli

X1

X2

ε11 = 2.3×10−12 ×104 = 2.3×10−8

Full derivation in Handouts

Examples of Piezoelectric Materials¡ Ammonium dihydrogen phosphate (ADP)¡ Used in sonar trasducers

¡ Fairly large d33 (more than 10 times the largest moduli in quartz)

¡ However, fairly soft and moisture sensitive.


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0 0 0 1.8 0 00 0 0 0 1.8 00 0 0 0 0 48

!

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###

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%

&&&×10−12C / N

Piezoelectric moduli

PZT (PbZr1−xTixO3)¡ Used extensively in trasducer technology¡ Underwater sonar, biomedical ultrasound, multilayer actuators for

fuel injection, piezoelectric printers, and bimorph pneumatic valves all make use of poled PZT ceramics.


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3m with [111] polar axis 4mm with [001] polar axis

Piezoelectric properties enhanced by choosing compositions close to morphotropicboundary between rhombohedral and tetragonal distortions.Heating above Curie point, the crystal structure becomes centrosymmetric, and all electric dipoles vanish. As the material is cooled in the presence of a sufficiently large electric field, the dipoles tend to align with the applied field, all together giving rise to a nonzero net polarization.

PZT, contd¡ Electrically poled materials belong to Curie symmetry ∞m, which has

piezoelectric tensors of the following form:


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0 0 0 0 d15 00 0 0 d15 0 0d31 d31 d33 0 0 0

!

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&&&&

¡ Intrinsic effects: Under a tensile stress parallel to the dipole moment, there is an enhancement of the polarization along [001] = Z3. When a tensile stress is applied perpendicular to the dipole, the polarization along Z3 (d33 and d31) is reduced. When the dipole is tilted by shear stress, charges appear on the side faces (d15).

¡ Extrinsic contributions: Doping with higher or lower-valent ions.¡ Nb5+ for Ti4+ gives soft PZT with easily movable domain walls and a large

piezoelectric coefficient. Used in hydrophones and other sensors. ¡ Fe3+ for Ti4+ gives hard PZTs, used in high-power transducers because they

will not depole under high fields in the reverse directions.

Elasticity¡ All materials undergo shape change under a mechanical force.

¡ Hooke’s Law (small stresses):

¡ where s are known as the elastic compliance coefficients (m2/N), and c are known as the stiffness coefficients (N/m2).

¡ Typical values:¡ Fairly stiff material like a metal or a ceramic, c is about 1011 N/m2

¡ Note that Hooke’s Law does not describe the elastic behavior at high stress levels that requires higher order elastic constants. Irreversible phenomena such as plasticity and fracture occur at still higher stress levels.


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εij = sijklσ kl

σ ij = cijklεkl

Stiffness and compliance coefficients¡ The stiffness and compliance coefficients are rank 4

tensors and transform as:

¡ There are 81 (34) coefficients in total, though the actual number of non-zero independent coefficients is considerably reduced by the fact that both the strain and stress are symmetric rank 2 tensors.


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cijkl! = aimajnakoalpcmnop

sijkl! = aimajnakoalpsmnop

Voigt notation¡ Similarly, the stiffness and compliance can be

represented as 6x6 matrices operation on the 6 element stress and strain vectors.

¡ However, the energy density is given by:

¡ Hence, the compliance and stiffness matrices comprises of at most 21 independent values. Symmetries above triclinic would have fewer independent values.


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dW =σ idεi = cijε jdεi∂2W∂ε j∂εi

= cij and ∂2W

∂εi∂ε j= cji

Since the order of differentiation does not matter,cij = cji

σ1σ 2

σ 3

σ 4

σ 5

σ 6

!

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&&&&&&&&

=

c11 c12 c13 c14 c15 c16c12 c22 c23 c24 c25 c26c13 c23 c33 c34 c35 c36c14 c24 c34 c44 c45 c46c15 c25 c35 c45 c55 c56c16 c26 c36 c46 c56 c66

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&&&&&&&&

ε1ε2ε3γ4γ5γ6

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&&&&&&&&

ε1ε2ε3γ4γ5γ6

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&&&&&&&&

=

s11 s12 s13 s14 s15 s16s12 s22 s23 s24 s25 s26s13 s23 s33 s34 s35 s36s14 s24 s34 s44 s45 s46s15 s25 s35 s45 s55 s56s16 s26 s36 s46 s56 s66

!

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&&&&&&&&

σ1σ 2

σ 3

σ 4

σ 5

σ 6

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%

&&&&&&&&

Conversion between tensor and matrix notations¡ Stiffness


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σ ij = cijklεklσ11 = c1111ε11 + c1112ε12 + c1121ε21 + c1122ε22 +…The corresponding matrix form isσ1 = c11ε1 + c16γ6 + c12ε2 +… (recall that γ6 = ε12 +ε21)Hence, there is no factor of 2 in the conversion for the stiffness tensor to matrix.

Conversion between tensor and matrix notations - Compliance


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ε1ε2ε3γ4γ5γ6

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&&&&&&&&

=

s1111 s1122 s1133 2s1123 2s1113 2s1112s1122 s2222 s2233 2s2223 2s2213 2s2212s1133 s2233 s3333 2s3323 2s3313 2s33122s1123 2s2223 2s3323 4s2323 4s2313 4s23122s1113 2s2213 2s3313 4s2313 4s1313 4s13122s1112 2s2212 2s3312 4s2312 4s1312 4s1212

!

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&&&&&&&&

σ1σ 2

σ 3

σ 4

σ 5

σ 6

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&&&&&&&&

Blackboard

Symmetry restrictions on compliance and stiffness coefficients

¡ Similar to piezoelectric coefficients, we need to use the full tensor notation to work out the effects on symmetry on the compliance and stiffness coefficients.

¡ Let’s consider a crystal with point group 4/m. Considering the 4 // Z3, we have X1’ = X2, X2’ = -X1, X3’ = X3. Hence, we can show that


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s11 s12 s13 0 0 s16s12 s11 s13 0 0 −s16s13 s13 s33 0 0 00 0 0 s44 0 00 0 0 0 s44 0s16 −s16 0 0 0 s66

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$$$$$$$$

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''''''''

Blackboard

Stiffness matrices for the 32 point groups


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Stiffness and compliance relations


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Using the matrix notation,σ = cε = csσHence, cs =1Converting to the Einstein notation,cijs jk = δikIn full tensor notation, we havecijklsklmn = δimδ jn

Example: For cubic crystals with only three independent coefficients,it can be shown that:

c11 =s11 + s12

(s11 − s12 )(s11 + 2s12 )

c12 = −s12

(s11 − s12 )(s11 + 2s12 )

c44 =1s44

Compressibility¡ Compressibility is defined as

¡ where V is volume and p is hydrostatic pressure.

¡ As noted earlier, change in volume per unit volume is:

¡ For hydrostatic pressure, we have


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K = −1VdVdp

ΔVV

= ε11 +ε22 +ε33 = εii

εii = siikl pδklHence,K = s1111 + s1122 + s1133 + s2211 + s2222 + s2233 + s3311 + s3322 + s3333

(sum of upper left 3x3 sub-matrix of compliance matrix)

s11 s12 s13 0 0 s16s12 s11 s13 0 0 −s16s13 s13 s33 0 0 00 0 0 s44 0 00 0 0 0 s44 0s16 −s16 0 0 0 s66

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''''''''

Typical compressibilities in rocksaltcrystals

¡Crystals with long weak bonds tend to have higher compressibilities.


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Anisotropy factor¡ Consider the stiffness in an arbitrary Z1’ direction. We

have:

¡ For cubic crystals, we only have three non-zero elements, which gives

¡ So even in cubic crystals, the stiffness is anisotropic. The only exception is when

¡ Hence, is defined as the anisotropy factor.


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c1111! = a1ma1na1oa1pcmnop

c11! = c11 − 2(c11 − c12 − 2c44 )(a11

2 a122 + a11

2 a132 + a12

2 a132 )

c11 = c12 + 2c44

A = 2c44c11 − c12

Typical values of A¡ A = 1 implies the crystal is isotropic.

¡ A < 1 implies that the crystal is stiffest along <100> cube axes

¡ A > 1 implies crystal is stiffest along <111> cube diagonals.


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Stiff along <111>

Stiff along <100>

Summary on Tensors¡ In this second part of the course, you have been given an overview of tensors

and their application to represent fields (e.g., stress, strain, polarization) and properties (e.g., conductivity, piezoelectric moduli, stiffness and compliance).

¡ It is of course impossible to cover all the different kinds of material properties, but the concepts and tools you have learned (transformation of tensors, symmetry restrictions on tensor values, etc.) thus far should be readily generalizable to any properties you encounter in future.

¡ Gap in coverage: Due to limits of time, I have elected not to cover magnetic group symmetries and axial tensor properties. Interested parties are referred to the relevant sections in Structure of Materials and Properties of Materials for a detailed discussion on these topics. The general concepts are similar, but the introduction of time reversal greatly increases the complexity of symmetry considerations (122 magnetic point groups and 1651 magnetic space groups!)


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UCSD NANO106 - 09 - Piezoelectricity and Elasticity

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