Lecture 5 Analysis and design of electro thermally actuated MEMS

Solving three sets of coupled partial differential equations and its implications in design.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.1

Contents

• Overview of thermal actuators • Principle of electro-thermal-compliant (ETC) actuation • Analysis issues – Thermal modeling • Design issues • Examples Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.2

Bimorph effect is used widely…

Heating is easily achieved in MEMS with Joule heating.

Mismatched thermal expansion coefficients of two materials make a bi-material structure bend.

Benecke, W. and Riethmuller, W, 1989, “Applications of Silicon Microactuators Based on Bimorph Structures,” Proceedings of the IEEE MEMS Workshop, Salt Lake City, Utah, Feb. 1989, pp. 116-120.

Takeshima, N. and Fujita, H, 1990, “Polyimide Bimorph Actuators for a Ciliary Motion system,” Micromechanical Sensors, Actuators, and Systems, 1991, pp. 203 209.

Chu, W.-H. and Mehregany, M., 1994, “MicrofabricatedTweezers with a Large Griping Force and a Large Range of Motion,” Technical Digest of Solid State Sensors and Actuators Workshop, Hilton Head Island, SC, June 1994, pp. 107-100.

The earliest analysis of this goes back to Timoshenko: Timoshenko, S., “Analysis of Bi-metal Thermostats,” J. of the Optical Society of America, Vol. 11, 1925, pp. 233-255.

Slide 5.3

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Electro-thermal actuation using a single material structure

Guckel, H., Klein, J., Christenson, T., Skrobis, K., Laudon, M., and Lovell, E.G., 1992, ‘Thermomagnetic Metal Flexure Actuators,” Technical Digest of Solid State Sensors and Actuators Workshop, Hilton-Head Island, SC, 1992, p 73.

Comtois, J. and Bright, V, 1996, “Surface Micromachined Polysilicon Thermal Actuator Arrays and Applications,” Technical Digest of Solid-State Sensor and Actuator Workshop, Hilton Head Island, SC, June 1996, pp. 174-177.

Lerch, P, Slimane, C.K., Romanwicz, B., and Renaud, P, 1996, “Modelization and characterization of asymmetrical thermal micro-actuators,” J. Micromechanics and Microengineering, Vol. 6, 1996, pp. 134-137 Moulton, T., 1997, “Analysis and design of Electro-Thermal-Compliant micro devices” Center for Sensor Technologies at the university of Pennsylvania technical report #TR-CST31DEC97, pp.13-26.

Keller, C.G. and Howe, R.T., 1997, “Hexsil Tweezers for Teleoperated Micro-Assembly,” Proc. 10th Annual International Workshop on Micro-Electro-Mechanical Systems (MEMS '97), Nagoya, Japan, January 26-30, 1997, pp. 72-77.

Pan, C. S. and Hsu, W., 1997, “An electro-thermally and laterally driven polysilicon microactuator,” J. Micromechanics and Microengineering, 7 (1997), pp. 7-13.

Sigmund, O., “Topology Optimization in Multiphysics Problems,” Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium, Vol. 3, St Louis, August 1998, pp. 1492-1500.

Cragun, R. and Howell, L.L., “A Constrained Thermal Expansion Micro-Actuator,” Proceedings of the Micro-Electro- Mechanical Systems (MEMS) Symposium at the International Mechanical Engineering Congress and Exhibition, DSC Vol. 66, pp. 365-371.

Huang, Q. and Lee, N., “Analysis and Design of Polysilicon Thermal Flexure Actuator,” J. Micromechics and Microengineering., Vol. 9, 1998, pp. 64-70.

Comtois, J. H., Michalicek, M.A., and Barron, C.C., “ Electrothermal actuators fabricated in four-level planarized surface micromachined polycrystalline silicon.” Sensors and Actuators A Physical, Vol. 70, 1998, pp 23-31.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.4

Electro-Thermal-Compliant MEMS

The structure is flexible.

They are small.

Joule heating causes thermal loads.

Electrical actuation by applying an voltage.

Slide 5.5

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Embedded ETC actuation

Actuator and mechanism are together.

Temperature Moulton, T. and Ananthasuresh, G.K., “Design and Manufacture of Electro-Thermal-Compliant Micro Devices,” Sensors and Actuators, Physical, 90 (2001), pp. 38-48.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.6

In series connection

(Guckel et al., 1992; Comtois and Bright, 1996) Bends up V Temperature Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.7

In parallel connection

(Moulton and Ananthasuresh, 1997) Bends down V Temperature Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.8

Prototype in the series mode

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.9

Prototype in the parallel mode

25 um diameter gold wire Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.10

Changing electrical resistivity with doping (if made with silicon)

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.11

Changing the length of the flexure

Bends downwards Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.12

ETC expansion building block

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.13

Parallel micro manipulator

With three degrees of freedom; Made using MUMPs, polysilcon.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.14

Analysis of ETC devices

Specified voltage Thermal flux Fixed V Traction force Specified temperature Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.15

Three analyses

Electrical

Joule heating term

Thermal

Voltage and current Temperature Temperature

Thermo-elastic

Deformation, stresses, and strains.

The equations are coupled because… -- almost all properties are temperature-dependent -- deformation can effect thermal boundary conditions (e.g., convection and radiation) Slide 5.16

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Electrical analysis: steady-state equilibrium equations

  

k e



V V



V

specified   on 0 in 

eE

 Strong form   

T V k e



V v d

  0 Weak form

V V v k e

= voltage = “virtual” voltage = electrical conductivity Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.17

Thermal analysis: Steady-state equilibrium equations

Joule heating 

T

n

 

k t



T

    

k T

specified

t



T

  

T

on

V k e



eT f nT

on 

V



nT

 0 in    Fixed temperature Convection and radiation Strong form   

T T k t



T v d

    

T V k e



V T v d

  0 Weak form

T T v k t

= Temperature = “virtual” temperature = thermal conductivity Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.18

Thermo-elastic analysis: static equilibrium equations

ε



σ

 

σ E

 :  0

ε

  (

T

 

T

 )

I

 ( 

u

 

T u

) / 2 Strong form

u



u

specified on 

eM

 

ε

(

u

) :

E

:

ε

(

u

v

)

d

   

ε

th

:

E

:

ε

(

u

v

)

d

  0 Weak form

ε σ u

k



t th

= stress

ε

= deformation = elastic constitutive properties   (

T



T

 )

I

= stress

u

v

= virtual deformation = thermal strain = thermal expansion coefficient

T

 = ambient temperature Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.19

Issues in thermal modeling

• Convection – Temperature dependence of heat transfer properties.

– Size dependence of heat transfer properties.

• Radiation – View / Shape factors. – Radiation heat transfer between parts of the same device which are at different temperatures.

• Boundary Conditions – Essential Boundary conditions at the device anchor.

– Natural Boundary conditions at the device anchor.

• Conduction through trapped air volume – Conduction between parts of the same device at different temperature with an intervening trapped air volume. – Conduction from the underside of the device to the substrate through the air trapped between them.

• Temperature dependence of thermo-physical Properties Slide 5.20

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Why is convection so important?

Thermal Expansion Device (TED), Cragun & Howell (1998)

Without

convection or radiation

With

convection and radiation Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.21

Essential vs. natural boundray conditions

Essential

boundary conditions

Thermally Grounded Natural

boundary conditions

Not Thermally Grounded

Silicon Device SiO 2 Silicon Handle Glass Ground Having one or the other makes a big difference.

Slide 5.22

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Scaling effects are quite significant

20 node, 3-D Continuum finite elements in ABAQUS

Fully Coupled

Electro-Thermal Analysis

Sequentially Coupled

Thermo-Elastic Analysis With temperature dependent material properties and heat transfer coefficients.

Mankame, N. and Ananthasuresh, G. K., “Comprehensive Thermal Modelling and Characterization of an Electro-Thermal Compliant Microactuator,” J. Micromechanics and Microengineering, Vol. 11, 2001, pp. 452-462.

Meso Micro For the same maximum temperature, meso (up to a cm) scale device provides more deflection.

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.23

Experimental validation of temperature distribution

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.24

Dealing with more complicated geometry…”line element” modeling

Beam 1 Encastre supports Beam 4 Beam 3 Thermo-elastic Model Beam 2 T in Narrow arm, seg. 1 End connection, seg. 2 R 1 T out R 4 R 3 R 2 Electrical Model Flexure, seg. 4 Wide arm, seg. 3 Thermal Model Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.25

Electro-thermal-compliant design

Three types of problems Uniform temperature rise Desired displacement Thermal insulation Mechanical constraint Objective Output disp.

Non-uniform temperature with external heating Desired displacement Heat flux Thermal insulation Mechanical constraint Outdisp & temp.

Non-uniform heating with voltage (Joule heating) V Desired displacement Thermal insulation Mechanical constraint Outdisp., temp. & current Slide 5.26

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

With multiple materials

V M 1 M 2 M n

x

Output displacement

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.27

Design parameterization

Material properties

   

E

  

k k



t e

       

m n

  1 exp    (   2   2

m m

) 2        

k k

E



t e m m m m

        

k e void k t void void E ijkl

   

void ij

    

Convection

h



m n

  1    exp    (    2  2

m m

) 2     

h

0 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.28

Interpolation of convection

h convection heat transfer coefficient Hole created during optimization

S k



h

( 1  

k

)

T

d

S k

  

S



k hT

0 d

S k

 

k k

 1  0 fixed Heat flux

e S k



k

 0 convection Output displacement Weak form of the thermal equilibrium equation:   

T T v k t



T

d     

e

 

k Ns

 1

S k T v h

( 1  

k

)(

T



T

 ) d

S k

      

V T k e



V



T v

d   0

Convection through element surfaces only if the neighboring elements are empty

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.29

Optimization problem

Minimize

ρ



u out

  

T V k e



V v d

  0   

T T v k t



T

d     

e

 

k Ns

 1

S k T v h

( 1  

k

)

T

d

S k

      

V T k e



V



T v

d   0  

ε

(

u

) :

E

:

ε

(

u

v

)

d

  

ε

th

:

E

:

ε

(

u

v

)

d

  0   ( )

d

 

V

*  0     

k k

E



t e

( ( ( (

x

)

x x x

) ) )      

m n

  1

S m

(

ρ

(

x

))  

k k t e m m

 

E



m m

    Slide 5.30

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Optimality criteria

First Variation of the Lagrangian = 0 

A

 

D

 0

D

 

S

  

m n

  1      

m

  2

m

 exp    (   2   2

m m

) 2  

A

  

u out

  ....

      

T V

ε

(

u

) 

k e

  : 

E

  

V v

:

ε

(

u

v

)

d



e d



e

    

T T

 

ε

th



k t

  : 

E

  

T v d



e

:

ε

(

u

v

) 

d



e e

 

k N s

    1

S k

   

ε

 

th



h

  ( 1  

k

)(

T

:

E

:

ε

(

u

v

)

d



e



T

 )

dS k

  ...

Adjoint volt.:

  

T V v k e

 

V

d   2   

T V k e

 

V T v

d   0

Adjoint temp.: Adjoint disp.:

  

T T v k t

 

T

d  

e

 

k N s

    1

S k h

( 1  

k

) 

T dS k

    

ε

(

u

v

) :

E

: 

I



T

d   

ε

( 

u

) :

E

:

ε

(

u

v

) d   

u out



u



u

 0  0 Solve from bottom to top Slide 5.31

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Variable update scheme

Lagrangian multiplier for volume constraint: 

B

 

V m n

  1   

B

(

k

)

d

 exp       (   

m

2  2

m D

(

k

)  (

k

)

d

   

l D

(

k

) 

l d

 

a



D

(

k

)

D

(

k

)

d

 ) 2    

u



D

(

k

) 

u d

 Solved in an inner loop  (

k

 1 )   (

k

)   (

A

(

k

)  

D

(

k

) )

k

is iteration number  (

k

) ( 1   )   (

k

 1 )   (

k

) ( 1   )  Step length  Move limit Slide 5.32

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Uniform heating with one material

Intuitive Non intuitive Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.33

Uniform heating with two materials

E

1  5

E

2  1  2  2 • Red color stiffer material • Blue color flexible material • only the top portion of the square design domain is utilized

E

1  5

E

2  1   2 / 2 • Since the objective is to maximize the downward displacement, it helps if the middle bar expands less Slide 5.34

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Non-uniform heating with heat flux input

Intermediate material to prevent heat flow to the output port

specifications Optimal topology

70 60 50 40 30 20 10

Initial temp. profile final temp. profile

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh 200 180 40 20 0 160 140 120 100 80 60 Slide 5.35

Effect of convection

No convection

160 140 120 100 80 60 40

Only top and bottom convection

350 300 250 200 150 100 50

Side surface as well as top and bottom convection

50 45 40 5 0 35 30 25 20 15 10 20 0 Temperature profiles Side convection has two effects on the topology optimization: to make boundary smoother to preserve heat; to make boundary rougher to dissipate heat.

Slide 5.36

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

ETC design example 1

Single material Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.37

ETC design example 2

Single material Two materials Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.38

ETC design example 3

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.39

Eliminating the “intermediate” material Penalty to be added to the objective to run optimization again…

f obj

(

new

) 

f obj

(

old

)   

e

 ( 1   ) Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.40

ETC design example 4

Single material Two materials Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.41

Microfabrication of structures with in-plane heterogeneity

Micro-structure with in-plane heterogeneity Thin deposited metal (seed layer) Electroplated metal Silicon Subtrate silicon After shallow maskless etching of metal and deep etching of the substrate from underneath Gold Side views First trial with Si and gold Slide 5.42

Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh

Main points

• Three analyses need to be done to simulate ETC devices • Thermal modeling is not trivial • Ideas from design in single energy domain easily extend to multiple energy domains – Adjoint method is powerful Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., 2003. G. K. Ananthasuresh Slide 5.43