Download Micromachined Piezoelectric Devices

Micromachined Piezoelectric Devices
Chang Liu
Micro Actuators, Sensors, Systems Group
University of Illinois at Urbana-Champaign
Chang Liu
MASS
UIUC
Definition
• Direct Piezo Effect
– a mechanical stress on a material produces an electrical
polarization
• Inverse Piezo Effect
– an applied electric field in a material produces dimensional
changes and stresses within a material.
• In general, both piezoelectricity and inverse piezoelectricity are
denoted piezoelectric effects.
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History
• First discovered in 1880 by Curies.
• Two important macro-scale applications that defined the growth
of field
– quartz resonator for timing standard
• The frequency of the quartz oscillator is determined by the cut and
shape of the quartz crystal.
• miniature encapsulated tuning forks which vibrate 32,768 times per
second
– ultrasonic transceivers for marine warfare and medical imaging.
Tuning fork
quartz resonator
Walter Cady (1874-1973)
Inventor of quartz resonator
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Submarine Applications
Civil/military
monitoring
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Asymmetric Crystal Produces Piezoelectric Effect
• Symmetric (centrosymmetric) lattice structure does not produce
piezoelectricity when deformed.
• Asymmetic lattice structures do!
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The Coordinate System
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Direct
•
•
•
•
•
D: Electrical Polarization
T: Applied Mechanical Stress
d: Piezoelectric Coefficient Matrix
ε: Electrical Permittivity Matrix
E: Electrical Field
D  dT  E
 D1   d11
 D   d
 2   21
 D3  d 31
Chang Liu
d12
d 22
d 32
d13
d 23
d 33
d14
d 24
d 34
d15
d 25
d 35
T1 
T 
d16   2   11  12
T3  

d 26      21  22
T4 


d 36 
 32
T5   31
 
T6 
 13   E1 
 23   E 2 
 33   E3 
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Inverse Piezoelectricity
• s: Strain Vector
• S: Compliance Matrix
s  ST  dE
 s1   S11
s  S
 2   21
 s3   S 31
 
 s 4   S 41
 s5   S 51
  
 s6   S 61
Chang Liu
S12
S 22
S 32
S 42
S 52
S 62
S13
S 23
S 33
S 43
S 53
S 63
S14
S 24
S 34
S 44
S 54
S 64
S15
S 25
S 35
S 45
S 55
S 65
S16  T1   d11




S 26  T2   d12
S 36  T3   d13
   
S 46  T4   d14
S 56  T5   d15
  
S 66  T6   d16
d 21
d 22
d 23
d 24
d 25
d 26
d 31 

d 32 
 E1 

d 33  
 E2
d 34   
 E3 

d 35 
d 36 
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Unit of Piezoelectric Coefficient
• Unit of d33 is the unit of electric displacement over the unit of
the stress. Thus
FV
[ D] [ ][ E ] m m Columb
[d 33 ] 



N
[T ]
[T ]
N
m2
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Reverse Piezoelectricity
• Strain as a function of applied field is governed by
0 d 31
 1   0
  

0 d 31
 2  0
 E1 
 3  0

0 d 33  
 
 E 2 
  4   0 d15 0  
E3 
  5   d15 0


0 
  
 6  0
0
0 
  
• Verify the unit
[ ] 1
C
C



[E] V C  (V ) N
m
m
– charge multiplied by electric field is force.
[d ] 
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Which is Axis 3?
• If no poling is applied, the axis normal to the substrate of
deposition is axis 3.
• If poling is applied, the axis of applied poling is axis 3.
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Semiconductors – Are they piezoelectric?
• Si is symmetric and does not exhibit piezoelectricity.
– (Si: positive charge; bond electrons: negative change)
• GaAs lattice is not symmetric and exhibits piezoelectricity.
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Common Piezoelectric Materials
• ZnO
– sputtered thin film
– d33=246 pC/N
• Lead zirconate titanate
(PZT)
– ceramic bulk, or sputtering
thin film
– d33=110 pC/N
• Quartz
– bulk single crystal
– d33=2.33 pC/N
• Polyvinylidene fluoride
(PVDF)
– polymer
– d33=1.59 pC/N.
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Diagram of a sputtering
system.
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Issues for Materials
• Poling
– establishment of preferred sensing direction
– application of electric field for long period of time after material is
formed
• Curie temperature
– temperature above which the piezoelectric property will be lost.
• Material purity
– the piezoelectric constant is sensitive to the composition of the
material and can be damaged by defects.
• Frequency response
– most materials have sufficient leakage and cannot “hold” a DC
force. The DC response is therefore not superior but can be
improved by materials deposition/preparation conditions.
• Bulk vs thin film
– bulk materials are easy to form but can not integrate with MEMS
or IC easily. Thin film materials are not as thick and overall
displacement is limited.
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Table 2: Properties of selected piezoelectric materials.
Relative
permitivity
(dielectric
constant)
Young’s
modulus
(GPa)
Density
(kg/m3)
Coupling
factor (k)
Curie
temperature
(oC)
8.5
210
5600
0.075
**
PZT-4
(PbZrTiO3)
1300-1475
48-135
7500
0.6
365
PZT-5A
(PbZrTiO3)
1730
48-135
7750
0.66
365
Quartz
(SiO2)
4.52
107
2650
0.09
**
Lithium
tantalate
(LiTaO3)
41
233
7640
0.51
350
Lithium
niobate
(LiNbO3)
44
245
4640
**
**
PVDF
13
3
1880
0.2
80
Material
ZnO
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Quartz
0
0 
12.77  1.79  1.22  4.5
 1.79 12.77  1.22 4.5

0
0


 1.22  1.22 9.6
0
0
0 
12
2
s
  10 m / N
4.5
0
20.04
0
0 
  4.5
 0
0
0
0
20.04  9 


0
0
0
 9 29.1
 0
0
0
 2.3 2.3 0  0.67
d   0
0 0
0
0.67 4.6  10 12 C / N
 0
0 0
0
0
0 
0
0 
4.52
 r   0 4.52 0 
 0
0
4.52
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PZT
• lead zirconate titanate (Pb(Zrx,Ti1-x)O3, or PZT)
• Pb(Zr0.40,Ti0.60)TiO3
0
0
0 293 0 
 0


d ij   0
0
0 293 0 0  pC / N
  44.2  44.2 117 0
0 0 

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ZnO
0
0
0
 11.34 0
 0
d   0
0
0
 11.34
0
0 pC / N
 5.43  5.43 11.37
0
0
0
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Bilayer Bending
• Ap and Ae are the cross-section areas of the piezoelectric and the
elastic layer, Ep and Ee are the Young’s modulus of the
piezoelectric and the elastic layer, and tp and te are the thickness
of the piezoelectric and the elastic layer
2 slong (t p  t e )( Ap E p Ae Ee )
1

r 4( E p I p  Ee I e )( Ap E p  Ae Ee )  ( Ap E p Ae Ee )(t p  t e ) 2
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Example 1
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3
Actuator Example
1
Cr/Au
Si3N4
ZnO
Si3N4
Cr
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Example 2
• A patch of ZnO thin film is located near the base of a cantilever
beam, as shown in the diagram below. The ZnO film is
vertically sandwiched between two conducting films. The
length of the entire beam is l. It consists of two segments – A
and B. Segment A is overlapped with the piezoelectric material
while segment B is not. The length of segments A and B are lA
and lB, respectively. If the device is used as a force sensor, find
the relationship between applied force F and the induced
voltage.
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 D1   d11 d12
 D   d
 2   21 d 22
 D3  d 31 d 32
d13
d14
d15
d 23
d 24
d 25
d 33
d 34
d 35
T1 
T 
d16   2   11  12  13   E1 
T3  

d 26      21  22  23   E2  
T4 


d 36 

 33   E3 
T5   31 32
 
T6 
0
0
0
 11.34 0
 0
d   0
0
0
 11.34
0
0 pC / N
 5.43  5.43 11.37
0
0
0
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Solution
The c-axis (axis 3) of deposited ZnO is generally normal to the front surface of the substrate
it is deposited on, in this case the beam. A transverse force would produce a
longitudinal tensile stress in the piezoelectric element (along axis 1), which in turn
produces an electric field and output voltage along the c-axis. The shear stress
components due to the force is ignored.
The stress along the length of the piezoresistor is actually not uniform and changes with
position. For simplicity, we assume the longitudinal stress is constant and equals the
maximum stress value at the base. The maximum stress induced along the longitudinal
direction of the cantilever is given by
 1,max  Mt /( 2 I )  Flt beam / 2 I beam
.
The stress component is parallel to axis 1.
According to Equation 2, the output electric polarization in the direction of axis 3 is
.
The overall output voltage is then
3
31 1, max
.
D3t piezo Flt beamt piezo
with Tpiezo being the thickness of the piezoelectric stack. V  E t

3 piezo 
D d 

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2I beam
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Example 3
• For the same cantilever as in Example 2, derive the vertical
displacement at the end of the beam if it was used as an
actuator. The applied voltage is V3.
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Under the applied voltage, the electrical field in axis 3 is
E3 
V3
t piezo
The applied electric field creates a longitudinal strain along axis 1, with the magnitude given
by Equation 5 as
S1  E3 d 31
Segment A is curved into an arc. The radius of the curvature r due to applied voltage can be
found from Equation 13.
 (x  l A )
The displacement at the end of segment A, , can be found by following similar procedure
used in Example 1. The angular displacement at the end of the piezoelectric patch is
(x  lA ) 
lA
r
The segment B does not curl and remains straight. The vertical displacement at the end of
the beam is
 ( x  l )   ( x  l A )  l B sin  ( x  l A )
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Example 4
• A ZnO thin film actuator on a cantilever is biased by co-planar
electrodes. The geometry of beams and piezoelectric patches is
identical as in Example 2. Find the output voltage under the
applied force. If the structure is used as an actuator, what are
the stress components when a voltage is applied across the
electrodes?
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The applied force generates two stress components – normal stress T1 and shear stress
T5. The output electric field is related to the stresses according to the formula for direct
effect of piezoelectricity
T1 
 D1   d11
 D   d
 2   21
 D3  d 31
d12
d 22
d 32
d13
d 23
d 33
d14
d 24
d 34
d15
d 25
d 35
T 
d16   2   11  12
T 
d 26   3    21  22
T
d 36   4   31  32
T5 
 
T6 
 13   E1 
 23   E 2 
 33   E3 
Because no external field is applied, the terms E1, E2, and E3 on the righthand side of
the above equation are zero. The formula can be simplified to the form
Therefore,
T1 
0
0
0
0
 11.34 0  
 D1   0
D    0
  0   10 12
0
0

11
.
34
0
0
2
  
 0 
 D3   5.43  5.43 11.37
0
0
0  
T5 
 
 0 
The output voltage is related to the polarization in axis-1,
D3  5.43  10 12  T1
D1  11.34  10 12  T5
V1 
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D1

 lA
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Let’s find the output stress when the device is used as an actuator. Suppose a voltage V is
applied across the longitudinal direction. Here we assume the spacing between the two
electrode is lA, hence the magnitude of the electric field is
V
E1 
lA
The applied electric field creates a longitudinal strain along axis 1. The strain is found by
 s1   s11
s  s
 2   21
 s3   s31
 
 s 4   s 41
 s5   s51
  
 s 6   s 61
s12
s 22
s32
s 42
s52
s 62
s13
s 23
s33
s 43
s53
s 63
s14
s 24
s34
s 44
s54
s 64
s15
s 25
s35
s 45
s55
s 65
s16  T1   d11

s 26  T2   d12
s36  T3   d13
   
s 46  T4   d14
s56  T5   d15
 
s 66  T6   d16
d 21
d 22
d 23
d 24
d 25
d 26
d 31 

d 32 
 E1 
d 33   
 E2
d 34   
 E3 
d 35   
d 36 
Since no external stresses are applied, we set T1 through T6 zero. The simplified formula for
strain is
0
 5.43 
 s1   0
0
No longitudinal strain components are
generated in this manner.
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
s   0
0
0

5
.
43


2
 
 
 E1 
 s3   0
0
0
11.37   
 0  10 12   
 
 11.34
0  
s4   0
0
 0 


 s5 
S 5 
 11.34
0
0 
  
 
0
0 
 0 
 s 6   0
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Example 5
• Derive the expression for the end displacement of piezoelectric
transducer configured similarly as Example 4, with the
difference that the electrodes are used to pole the ZnO material.
In other words, axis-3 is now forced to lie in the longitudinal
direction of the beam length. A voltage V is applied across two
electrodes.
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The electric field in the longitudinal axis is
V
E3 
lA
The applied field induced a longitudinal strain (S3) according to
or
s3  d 33 E3
0
 5.43 
 s1   0


s 
0
0

5
.
43


 2
0
 s3   0
0
11.37   
 0  10 12
 
 11.34
0  
s4   0
  E3 
 s5    11.34
0
0

  
0
0 
 s 6   0
We should use s3 to replace slong in Equation 8. The subsequent analysis is similar to the one
performed for Example 3.
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Case 7.1: Acceleration Sensor
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Case 7.2: Membrane Piezoelectric Accelerometer
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Case 7.3: Piezoelectric Acoustic Sensor (Pressure
Sensor)
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Case 7.4: Piezoelectric Microphone
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Surface Elastic Waves
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