EE C245 - ME C218
Introduction to MEMS Design
Fall 2003
Roger Howe and Thara Srinivasan
Lecture 13
Alternative Transduction Principles
EE C245 – ME C218 Fall 2003 Lecture 13
Today’s Lecture
•
•
•
•
Piezoelectric materials for MEMS: courtesy of
Justin Black (jblack@eecs) and Prof. R. M. White
Piezoresistive strain sensing in silicon:
mechanism and device application
Thermal actuation: microtweezers
Reading:
Senturia, S. D., Microsystem Design, Kluwer
Academic Publishers, 2001, Chapter 18, pp. 470477, Chapter 21, 570-578.
EE C245 – ME C218 Fall 2003 Lecture 13
2
1
Origin of the Piezoelectric Effect
Several views of an α -quartz crystal
Z
X2
X3
Si
Y
O
X1
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
3
Origin of the Piezoelectric Effect
For r >> a, the electric field at the point P is:
E
p
= E
+
+ E
−
≈
3q
4 πε r
2
−
3q
4 πε r
2
= 0
The potential and electric field appear as if the charges are coincident at their
center of gravity (point O)
Ep ≅ 0
a/ 4
P
a/ 2
O
r >> a
Si atom
O atom
Introduction to Quartz Crystal Unit Design , Virgil Bottom, 1982.
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
4
2
Origin of the Piezoelectric Effect
♦
♦
♦
Assume the applied force F causes the
line OD to rotate counter clockwise by a
small angle dθ
This strain shifts the center of gravity of
the three positive and negative charges to
the left and right, respectively
60°
dθ
O
A dipole moment, p = qr, is created which
has an arm (r) of:
p = qr @ qa33/2 dθ
♦
F
D
Assuming the crystal contains N such
molecules per unit volume, each subject to
the same strain dθ, the polarization (or
dipole moment per unit volume) is:
P = Nqa33/2 dθ
polarization
strain
EE C245 – ME C218 Fall 2003 Lecture 13
P = Nqr
F
Si atom
O atom
J. Black and R. M. White
5
Origins of the Piezoelectric Effect
♦
For sufficiently small deformations, the polarization (P) is linearly related to the strain
(S) by:
P = gS
where g is the piezoelectric voltage coefficient.
♦
The polarization P equals the surface charge per unit area, or piezoelectric
displacement.
Converse Piezoelectric Effect
♦
When a piezoelectric crystal is placed in an electric field, pos itive and negative ions are
pushed in opposite directions and a dipole tends to rotate to align itself with the electric
field.
♦
The resulting motion gives rise to a strain S that is proportional to the electric field E
S = dE
where d is the piezoelectric charge coefficient.
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
6
3
Anisotropic Crystal Properties: Generalized Stress-Strain
Perspective and cross sectional
views of α -quartz
♦
In anisotropic materials a tensile
stress can produce both axial and
shear strain.
♦
For example, a thin, X-cut rod of
quartz subject to a tensile force will
not only become longer and thinner,
but it will also rotate about its
longitudinal axis.
♦
Since we have 6 components of
stress (T) and 6 components of strain
(S), 36 constants must be used to
describe behavior in the general
case.
♦
Crystal symmetry (e.g. trigonal,
hexagonal) greatly reduces the
number of independent constants.
Si
EE C245 – ME C218 Fall 2003 Lecture 13
O
7
J. Black and R. M. White
Anisotropic Crystal Properties: Generalized Stress-Strain
For small deformations, stress (T) and strain (S)
are related through the compliance matrix (s)
Sxx 
S 
 yy 
Szz 
 =
Syz 
S 
 zx 
Sxy 
 
s11
s
 21
s31

s41
s51

s61
s12
s22
s13 s14 s15
s23 s 24 s25
s32
s42
s52
s33 s34 s35
s43 s 44 s45
s53 s54 s55
s62
s63 s 64 s65
s16  Txx 
 
s26  Tyy 
s36  Tzz 
 
s46  Tyz 
s56  Tzx 
 
s66  Txy 
Quartz has threefold symmetry, physical
properties repeat every 120°.
Quartz is also symmetric about the X-axis
Z
Conservation of energy requires sij = sji. Performing
rotations based upon trigonal symmetry considerations, the
compliance matrix reduces to 6 independent coefficients:
 Sxx   s1 1 s1 2
  
 Syy   s 2 1 s1 1
 Szz   s
s1 3
  =  13
 Syz   s1 4 − s1 4
  0
0
 Szx  
 Sxy   0
0
s1 3 s1 4
s1 3 − s1 4
s3 3
0
0
s44
0
0
0
0
0
0
0
0
s4 4
2s1 4
 Txx 
 T 
  yy 
 Tzz 
 
 T yz 
2s1 4  Tzx 
 
2(s1 1 − s1 2 ) Txy 
EE C245 – ME C218 Fall 2003 Lecture 13
0
0
0
0
X2
X3
Y
X1
J. Black and R. M. White
8
4
Anisotropic Crystal Properties: Piezoelectric Constants
Recall that the strain (S) is related to the electric field (E) by the piezoelectric
charge coefficient matrix (d)
Sxx 
S 
 yy 
Szz 
 =
Syz 
S 
 zx 
Sxy 
 
d11 d21
d
 12 d 22
d13 d23

d14 d 24
d15 d25

d16 d 26
d31 
d32 
E x 
d33   
 E y 
d34 
E 
d35   z 

d36 
Applying the symmetry conditions for quartz, the piezoelectric strain matrix (d)
simplifies to:
STRAIN (S)
S xx 
S 
 yy 
Szz 
 =
Syz 
S 
 zx 
S xy 
 
 d11
0
− d
0
11

 0
0

0
 d14
 0
− d14

− 2d11
 0
0
0
E x 
0  
 E y 
0
E 
0  z 

0
FIELD (E)
X
EXTENSIONAL
Y
Z
X
Y
Z
SHEAR
(about axis)
X
Y
Z
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
9
Anisotropic Crystal Properties
Modes in Quartz
♦
Elastic modulus and compliance
♦
Thermal conductivity
♦
Electrical conductivity
♦
Coefficient of thermal expansion
♦
Dielectric constants
♦
Piezoelectric constants
♦
Optical index of refraction
♦
Velocity of propagation of
longitudinal waves
♦
Velocity of propagation of shear
waves
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
10
5
Constitutive Equations for Piezoelectric Materials
compliance
piezoelectric strain
coefficients
S = sE T + d E
strain
stress
electric field
piezoelectric strain
coefficients (transpose)
dielectric
permittivity
D = dTr T + ε T E
electric
stress
displacement
electric field
Superscripted material constants (e.g. s E ) are those values obtained when the
superscripted quantity is held constant.
EE C245 – ME C218 Fall 2003 Lecture 13
11
J. Black and R. M. White
Properties of Common Piezoelectrics
Stiffness
(1010 N / m2)
Strain Coefficient
(10-1 2 C / N)
Relative
Permittivity
Coupling
Coefficient K2
(%)
Velocity
(m / s)
Density
(kg / m3)
Aluminum Nitride
(AlN)
33.0
5.6 (d33)
8.6
6.0
11,300
3.26
Barium Titanate
(BaTiO 3) *
11.0 - 27.5
82-145 (d33)
625-1350
39 – 46
4460
5.85
Lithium Niobate
(LiNbO 3)
24.5
19.2 (d33)
44
17.2
4379 †
4.64
Lithium Tantalate
(LiTaO 3)
23.3
8.0 (d33)
41
4.7
4112 †
7.64
P(VDF –TrFE)
0.3
-12.0 (d31)
13
Quartz
(SiO 2)
10.7
2.3 (d11)
4.5
0.11
PZT
(PbZrTiO3)*
4.8 – 13.5
240-550 (d3 3)
1100- 3200
Zinc Oxide
(ZnO)
21.0
10-12 (d 33)
8.5
SAW Substrates
Thickness mode, thin film
Ferroelectric ceramic,
bulk material
Ferroelectric polymer
EE C245 – ME C218 Fall 2003 Lecture 13
†
†
0.18
k2 =
2400
1.88
3948 †
2.65
66 - 73
4600
7.55
7.5
6,080
5.60
†
mechanical energy stored
electrical energy applied
12
6
SAW Devices
Uz
♦
The stress-free boundary condition imposed
by the surface of a crystal gives rise to an
acoustic mode known as a surface acoustic
wave (Rayleigh wave)
♦
SAW energy is confined to within one
wavelength of the surface
♦
The components of surface particle motion,
Ux and Uz, are 90º out of phase, and Uz >>
Ux
♦
SAWs can be excited by interdigitated
transducers (IDTs) patterned on the surface
of piezoelectric crystals*
♦
IDT geometry allows construction of delay
lines, convolvers, correlators, pulse
compressors, filters
Ux
Used by the billions in communications and electronic
systems; invented by Prof. R. M. White, EECS Dept.,
UC Berkeley, 1965.
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
13
SAW State-of-the-Art
♦
Smallest SAW RF filter is 1.4 x 2.0 x 0.7 mm3 (EPCOS)
♦
Trend towards integration of passive components into ceramic SAW package
2 mm x 2mm x 0.8 mm3 SAW
from EPCOS
Integrated SAW duplexer:
receiver filter, transmit filter,
antenna matching network
Robert Weigel, et al ., “Microwave Acoustic Materials, Devices,and Applications,” IEEE Transactions on
Microwave Theory and Techniques, March 2002.
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
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7
Thin Film Bulk Acoustic Resonators
Piezoelectric thin film (ZnO
or AlN) sandwiched
between electrodes
Quarter wavelength
reflectors (e.g. W
and SiO2)
Piezoelectric thin film (ZnO
or AlN) sandwiched
between electrodes
Substrate
Solidly Mounted Resonator (SMR):
Quarter wavelength reflectors (also called
acoustic Bragg reflectors) confine
acoustic energy to top piezoelectric layer
FBAR: Remove substrate to confine
energy within the membrane.
Resonant frequency can range from 500 MHz to 20+ GHz.
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
15
Thin Film Bulk Acoustic Resonators
♦
Commercially available from Agilent
(FBAR) and TFR Technologies (SMR)
♦
Agilent’s volume is ~ 2.5 million / month
(Rich Ruby, Agilent, Oct. 2003)
♦
Advantages include:
♦
•
Up to 20 times area reduction
•
Lower parasitics
•
Steeper skirts
•
Lower insertion loss
•
Operation above 10 GHz
•
Power handling
www.agilent.com
Filter bandwidth limited by K2 of
piezoelectric film
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
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8
Derivation of Equivalent BVD Circuit
Assume piezoid is a thin plate of thickness d with infinite extent in x and z (1-d solution)
The equation of motion for the particle displacement ψ(y, t) is:
ρ
∂ 2ψ
∂ 2ψ
∂ψ
−c
+r
= 0
2
∂t
∂y 2
∂t
k = β − jα =
→ ψ ( y , t ) = A ( y )e j ω t
V (t ) = V m e jω t
ρω 2
ωr
− j
c
c
P = strain * g =
d
piezoid
y
x
c = stiffness
∂ψ
*g
∂y
ρ = density
r = damping
∂ 2V
σ
1 ∂P
= − =
∂y 2
ε
ε ∂y
i =ε
y = plate position
g = piezoelectric
coefficient
∂E
= i dielectric + i piezoelectric = Y ( jω )Vm e jωt
∂t
displacement
displacement
P = polarization
V = voltage
E = electric field
σ = charge density
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
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Equivalent BVD Circuit
K.M. Lakin, Thin Film Resonators and Filters, IEEE Ultrasonics Symp., Oct. 1999.
R=
d 3r
8Ag 2
Co =
Aε
d
ρd 3
L =
8 Ag 2
C =
Ra
Co
La
Ca
8 Ag 2 8 k 2 C o
=
π 2 dc
π2
k2 =
g2
cε
♦
Impedance governed by transducer area
♦
Typical SMR values are Co= 1.91 pF, Ca = 0.80 pF, Ra = 1.14 Ω, and La = 123.6
nH. These values correspond to a Q of approximately 1000†
EE C245 – ME C218 Fall 2003 Lecture 13
J. Black and R. M. White
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9
Piezoresistivity
• Strain dependence of electrical resistivity: a strong
effect in semiconductors, including silicon and poly-Si
• General expression: tensors galore!
resistivity
piezoresitivity tensor
(4th rank)
electric field
E = (ρ + Π σ) J
electric
field
Stress tensor
(2nd rank)
• Silicon: resistivity is a diagonal tensor with all
elements equal to ρ ; the stress tensor collapses with
cubic symmetry to a six-element array (reduced
notation)
EE C245 – ME C218 Fall 2003 Lecture 13
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Piezoresistive Coefficients in Silicon
•
Field-current equation has three independent piezoresistive
coefficients; along the <100> direction
E1 = ρ{[1 + π 11σ 1 + π 12 (σ 2 + σ 3 )] J1 + π 44 (τ 12 J 2 + τ 13 J 3 )}
•
The conventional π coefficients are related to the tensor
elements by:
Π 1111 = ρπ 11
Π 1122 = ρπ 12
Π 2323 = ρπ 44 / 2
EE C245 – ME C218 Fall 2003 Lecture 13
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10
Typical Embedded
Strain-Sensing Resistor
• Implantation defines a shallowresistor of opposite
type to bulk silicon structure
F
side view
X
top view
F
Orientation of piezoresistor
is longitudinal (aligned with
bending strain in beam)
EE C245 – ME C218 Fall 2003 Lecture 13
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Longitudinal and Transverse
Piezoresistive Coefficients
•
Relative resistance change can be expressed by the
longitudinal πl and transverse πt piezoresistive coefficients
∆R / R = π lσ l + π tσ t
•
Piezoresistors are often aligned to the wafer flat of (100) wafers,
which is in the [110] direction. Senturia, p. 473 provides the
result of coordinate transformations:
1
(π11 + π 12 + π 44 )
2t
1
π t ,110 = (π 11 + π 12 − π 44 )
2t
π l ,110 =
EE C245 – ME C218 Fall 2003 Lecture 13
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11
Silicon Piezoresistive Coefficients
• Function of type, doping, and temperature
ρ
π11
π12
π44
n-type
11.7
-102.2
53.4
-13.6
p-type
7.8
6.6
-1.1
138.1
Units: ρ [Ω-cm], π’s 10-11 Pa-1 Values are at T = 25oC
• Longitudinal and transverse coefficients in [110] direction
n-type:
π l = -31.2 x 10-11 Pa-1
π t = -17.6 x 10-11 Pa-1
p-type:
π l = 71.8 x 10-11 Pa-1
π t = - 66.2 x 10-11 Pa-1
EE C245 – ME C218 Fall 2003 Lecture 13
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Piezoresistor Placement
• Bulk micromachined diaphragm pressure sensor
R1
R3
R2
R1
R3
R4
EE C245 – ME C218 Fall 2003 Lecture 13
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12
Piezoresistive Wheatstone Bridge
•
Resistors experience both longitudinal and transverse stress
when diaphragm deflects: effects are almost equal and opposite
on two sets of resistors
∆R1 / R1 = π lσ l + π tνσ l = l (π l + νπ t )σ l = 67.6 x10− 11σ l
∆R2 / R2 = π tσ l + π lνσ l = (π t + νπ l )σ l = −61.7 x10− 11σ l
VDC
R2
R3
v
+ out -
R1
R4
EE C245 – ME C218 Fall 2003 Lecture 13
25
Piezoresistive Strain Sensing Summary
• Attractive for single crystal silicon structures, since
π’s are specified by crystal orientation
• Large literature on resistor sizing and placement for
minimizing sensitivity to lithographic errors
• Can result in highly sensitive transduction, depending
on design of mechanical structure and placement of
strain sensors
• No need for a Ph.D. in analog IC design from
Berkeley in order to amplify the output voltage from a
Wheatstone bridge! (Why? Impedance levels are
much lower … off-chip amplifier actually works.)
EE C245 – ME C218 Fall 2003 Lecture 13
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13
Limits of Piezoresistive Force Sensing
•
•
Scale AFM cantilever to 70 nm thickness; develop very shallow (20
nm) implanted piezoresistor. Tom Kenny, Stanford ME Dept.,
Hilton Head Workshop, June 2002.
Sensitivity is about 1 femto-Newton in a 1 Hz BW at 1 kHz
EE C245 – ME C218 Fall 2003 Lecture 13
27
Piezoresistors as “Internal Transducers”
•
Instrumented micro force plate: capacitive transduction would lead
to exposed electrodes, which could easily become contaminated
•
Application: measure forces exerted by a cockroach while walking
on all six legs … and while running on two legs!
Tom Kenny, Hilton Head Workshop, June 2002.
EE C245 – ME C218 Fall 2003 Lecture 13
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14
Measurements from Micro Force Plate
Prof. Robert Full, Dept. of Integrative Biology, UC Berkeley
Prof. Tom Kenny, ME Dept., Stanford
EE C245 – ME C218 Fall 2003 Lecture 13
29
Thermal Actuation
•
Simple idea: thermal strain à dimensional change
à generate desired motion. No tensors!
•
Challenges:
1. Thermal actuation requires static power dissipation
(compare electrostatic actuation)
2. “Heatuator” on EE 245 MUMPS chip: limits to using
geometry to control thermal expansion
3. Bandwidth of actuator is limited by thermal time constant …
but could have BW > 100 kHz for MEMS
EE C245 – ME C218 Fall 2003 Lecture 13
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15
Hexsil Tweezer
tweezer tips
thermal expansion bar
Linkage to amplify bar’s length
change to control tweezer tip
separation
Chris Keller, Ph.D. MSE Dept., UC Berkeley, 1998
EE C245 – ME C218 Fall 2003 Lecture 13
31
Basic Hexsil Fabrication Process
fabricate silicon mold
conformal sacrificial
layer deposition
conformal structural
layer deposition
Chris Keller, Ph.D. MSE Dept., UC Berkeley, 1998
EE C245 – ME C218 Fall 2003 Lecture 13
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16
Basic Hexsil Fabrication Process (cont.)
Hexsil structure (surface layer
holds molded portions together)
reuse mold
Deposit and pattern
surface layers (e.g., for
tweezer tips)
Release structure by means
of a (lengthy) sacrificial layer etch
Chris Keller, Ph.D. MSE Dept., UC Berkeley, 1998
EE C245 – ME C218 Fall 2003 Lecture 13
33
PolySi Hexsil Structures
Close-up of top of hexsil tweezer
leverage mechanism, showing
surface poly-Si face sheet
EE C245 – ME C218 Fall 2003 Lecture 13
Close-up of bottom of hexsil tweezer
leverage mechanism, showing
molded poly-Si ribs
34
17
Thermal Actuation of Tweezer Tips
50 mW dissispated in the thermal expansion bar will actuate
tweezer tips by about 40 µm; structure will withstand
temperatures up to 1100o C for periods of 10 minutes
Chris Keller, Ph.D. MSE Dept., UC Berkeley, 1998
EE C245 – ME C218 Fall 2003 Lecture 13
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