ee c245 - me c218 introduction to mems design fall … ee c245 – me c218 fall 2003 lecture 13 ee...

1

EE C245 – ME C218 Fall 2003 Lecture 13

EE C245 - ME C218Introduction to MEMS Design

Fall 2003

Roger Howe and Thara SrinivasanLecture 13

Alternative Transduction Principles

2EE 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, KluwerAcademic Publishers, 2001, Chapter 18, pp. 470-477, Chapter 21, 570-578.

2


Origin of the Piezoelectric Effect

Several views of an α -quartz crystal

Si OX1

X2

X3

Z

Y

J. Black and R. M. White



Si atom

O atom

a / 4 a / 2

O

P

r >> a

For r >> a, the electric field at the point P is:

The potential and electric field appear as if the charges are coincident at their center of gravity (point O)

04

34

322 =−≈+= −+ r

qr

qEEE p πεπε

Ep ≅ 0

Introduction to Quartz Crystal Unit Design , Virgil Bottom, 1982.


3



♦ 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

♦ A dipole moment, p = qr, is created which has an arm (r) of:

p = qr @ qa33/2 dθ

F

F

60°dθ

P = Nqr

O

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

strainpolarization

Si atom

O atom



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.


4


Anisotropic Crystal Properties: Generalized Stress-Strain

♦ 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.

Perspective and cross sectional views of α -quartz

Si O



X1

X2

X3

Z

Y

Anisotropic Crystal Properties: Generalized Stress-Strain

=

xy

zx

yz

zz

yy

xx

xy

zx

yz

zz

yy

xx

T

T

T

T

T

T

ssssss

ssssssssssssssssss

ssssssssssss

S

S

S

S

S

S

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211Quartz has threefold symmetry, physical properties repeat every 120°.

Quartz is also symmetric about the X-axis

Conservation of energy requires sij = sji. Performing rotations based upon trigonal symmetry considerations, the compliance matrix reduces to 6 independent coefficients:

For small deformations, stress (T) and strain (S) are related through the compliance matrix (s)

−

−

−

=

xy

zx

yz

zz

yy

xx

xy

zx

yz

zz

yy

xx

TT

TT

TT

sssss

ssssss

ssssssss

SS

SS

SS

)(22000020000

0000000000

121114

1444

441414

331313

14131121

14131211


5


Anisotropic Crystal Properties: Piezoelectric Constants

Recall that the strain (S) is related to the electric field (E) by the piezoelectric charge coefficient matrix (d)

=

z

y

x

xy

zx

yz

zz

yy

xx

E

E

E

ddd

ddddddddd

dddddd

S

S

S

S

S

S

362616

352515

342414

332313

322212

312111

Applying the symmetry conditions for quartz, the piezoelectric strain matrix (d) simplifies to:

−

−

−

=

z

y

x

xy

zx

yz

zz

yy

xx

E

E

E

d

dd

dd

S

S

S

S

S

S

020

0000000

0000

11

14

14

11

11

Z

Y

XEXTENSIONAL

Z

Y

XSHEAR(about axis)

ZYX

FIELD (E)STRAIN (S)



Anisotropic Crystal Properties

♦ 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

Modes in Quartz


6


Constitutive Equations for Piezoelectric Materials

D = dTr T + εT E

S = sE T + d Estrain stress electric field

piezoelectric strain coefficientscompliance

electric displacement

stress electric field

piezoelectric strain coefficients (transpose)

dielectric permittivity

Superscripted material constants (e.g. sE) are those values obtained when the superscripted quantity is held constant.



6,080

4600

3948 †

2400

4112 †

4379 †

4460

11,300

Velocity(m / s)

5.60

7.55

2.65

1.88

7.64

4.64

5.85

3.26

Density(kg / m3)

0.1813-12.0 (d31)0.3P(VDF–TrFE)

6.08.65.6 (d33)33.0Aluminum Nitride (AlN)

7.5

66 - 73

0.11 †

4.7 †

17.2 †

39 – 46

Coupling Coefficient K2

(%)

8.510-12 (d 33)21.0Zinc Oxide (ZnO)

1100- 3200240-550 (d3 3)4.8 – 13.5PZT(PbZrTiO3)*

4.52.3 (d11)10.7Quartz(SiO 2)

418.0 (d33)23.3Lithium Tantalate(LiTaO 3)

4419.2 (d33)24.5Lithium Niobate(LiNbO3)

625-135082-145 (d33)11.0 - 27.5Barium Titanate (BaTiO 3) *

Relative Permittivity

Strain Coefficient(10-1 2 C / N)

Stiffness(1010 N / m2)

Properties of Common Piezoelectrics

Thickness mode, thin film

Ferroelectric ceramic, bulk material

SAW Substrates

Ferroelectric polymer appliedenergyelectricalstoredenergymechanicalk =2

7


SAW Devices

♦ 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

Uz


Used by the billions in communications and electronicsystems; invented by Prof. R. M. White, EECS Dept.,UC Berkeley, 1965.


SAW State-of-the-Art

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.

♦ Smallest SAW RF filter is 1.4 x 2.0 x 0.7 mm3 (EPCOS)

♦ Trend towards integration of passive components into ceramic SAWpackage


8


Thin Film Bulk Acoustic Resonators

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.

Quarter wavelength reflectors (e.g. W and SiO2)

Substrate

Piezoelectric thin film (ZnO or AlN) sandwiched between electrodes

Piezoelectric thin film (ZnO or AlN) sandwiched between electrodes

Resonant frequency can range from 500 MHz to 20+ GHz.



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

♦ Filter bandwidth limited by K2 of piezoelectric film

www.agilent.com


9


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:

02

2

2

2

=∂

∂+

∂∂

−∂∂

tr

yc

tψψψ

ρ

cr

jc

jkωρω

αβ −=−=2

tjeyAty ωψ )(),( =→

c = stiffness

ρ = density

r = damping

y = plate position

g = piezoelectric coefficient

P = polarization

V = voltage

E = electric field

σ = charge density

gy

gstrainP **∂∂

==ψ

yP

yV

∂∂

=−=∂∂

εεσ 1

2

2

tjm

ntdisplacemericpiezoelect

ntdisplacemedielectric eVjYii

tE

i ωωε )(=+=∂∂

=

tjm eVtV ω=)(

piezoidd y

x



2

3

8 AgdL ρ=

2

3

8Agrd

R =

2

2

2

2 88ππ

oCkdc

AgC ==

dA

C oε

=

Equivalent BVD Circuit

♦ 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†

K.M. Lakin, Thin Film Resonators and Filters, IEEE Ultrasonics Symp., Oct. 1999.

Co

Ra

La

Ca

εcg

k2

2 =


10


Piezoresistivity

• 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)

E = (ρ + Π σ) Jelectric field

Stress tensor (2nd rank)

electric field

piezoresitivity tensor(4th rank)resistivity

• Strain dependence of electrical resistivity: a strong effect in semiconductors, including silicon and poly-Si

• General expression: tensors galore!


Piezoresistive Coefficients in Silicon• Field-current equation has three independent piezoresistive

coefficients; along the <100> direction

( ) )(]1[ 31321244132121111 JJJE ττπσσπσπρ +++++=

• The conventional π coefficients are related to the tensor elements by:

111111 ρπ=Π

121122 ρπ=Π2/442323

ρπ=Π

11


Typical EmbeddedStrain-Sensing Resistor

• Implantation defines a shallowresistor of opposite type to bulk silicon structure

F

Ftop view

side view

X

Orientation of piezoresistoris longitudinal (aligned withbending strain in beam)


Longitudinal and Transverse Piezoresistive Coefficients

• Relative resistance change can be expressed by the longitudinal πl and transverse πt piezoresistive coefficients

ttllRR σπσπ +=∆ /

• 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:

)(21

441211110, ππππ ++=t

l

)(21

441211110, ππππ −+=t

t

12


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


Piezoresistor Placement• Bulk micromachined diaphragm pressure sensor

R1 R3

R1

R4

R3

R2

13


Piezoresistive Wheatstone Bridge• Resistors experience both longitudinal and transverse stress

when diaphragm deflects: effects are almost equal and opposite on two sets of resistors

lltllltll xRR σσνππνσπσπ 1111 106.67)(/ −=+=+=∆

( ) llltlllt xRR σσνππνσπσπ 1122 107.61/ −−=+=+=∆

R4R1

R2 R3

+ -vout

VDC


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.)

14


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


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.

15


Measurements from Micro Force Plate

Prof. Robert Full, Dept. of Integrative Biology, UC BerkeleyProf. Tom Kenny, ME Dept., Stanford


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 expansion3. Bandwidth of actuator is limited by thermal time constant …

but could have BW > 100 kHz for MEMS

16


Hexsil Tweezer

Chris Keller, Ph.D. MSE Dept., UC Berkeley, 1998

thermal expansion bar

Linkage to amplify bar’s lengthchange to control tweezer tipseparation

tweezer tips


Basic Hexsil Fabrication Process

fabricate silicon mold conformal sacrificiallayer deposition

conformal structurallayer deposition


17


Basic Hexsil Fabrication Process (cont.)

Deposit and patternsurface layers (e.g., fortweezer tips)

Release structure by meansof a (lengthy) sacrificial layer etch

Hexsil structure (surface layerholds molded portions together)

reuse mold



PolySi Hexsil Structures

Close-up of top of hexsil tweezerleverage mechanism, showing

surface poly-Si face sheet

Close-up of bottom of hexsil tweezerleverage mechanism, showing molded poly-Si ribs

18


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 1100oC for periods of 10 minutes

ee c245 - me c218 introduction to mems design fall … ee c245 – me c218 fall 2003 lecture 13 ee...

Documents