1. Mechanics of Micro Structures
Micro Actuators, Sensors, Systems

2. Single crystal silicon and wafers
To use Si as a substrate material, it should be pure Si in a single crystal form
The Czochralski (CZ) method: A seed crystal is attached at the tip of a puller, which slowly pulls up to form a larger crystal
100 mm (4 in) diameter x 500 mm thick
150 mm (6 in) diameter x 750 mm thick
200 mm (8 in) diameter x 1000 mm thick

3. Miller indices
A popular method of designating crystal planes (hkm) and orientations <hkm>
Identify the axial intercepts
Take reciprocal
Clear fractions (not taking lowest integers)
Enclose the number with ( ) : no comma
<hkm> designate the direction normal to the plane (hkm)
(100), (110), (111)

4. Stress and Strain Definition of Stress and Strain
The normal stress (Pa, N/m2)
The strain
Poisson’s ratio

5. Hooke’s Law E: Modulus of Elasticity, Young’s Modulus The shear stress
The shear strain
The shear modulus of elasticity
The relationship

6. General Relation Between Tensile Stress and Strain

7. The behavior of brittle materials (Si) and soft rubber used extensively in MEMS
A material is strong if it has high yield strength or ultimate strength. Si is even stronger than stainless steel
Ductility is a measure of the degree of plastic deformation that has been sustained at the point of fracture
Toughness is a mechanical measure of the material’s ability to absorb energy up to fracture (strength + ductility)
Resilience is the capacity of a material to absorb energy when it is deformed elastically, then to have this energy recovered upon unloading

8. General Stress-Strain Relations
C: stiffness matrix
S: compliance matrix
For many materials of interest to MEMS, the stiffness can be simplified

9. Flexural Beam Bending Types of Beams; Fig. 3.15
Possible Boundary Conditions

10. Longitudinal Strain Under Pure Bending
Pure Bending; The moment is constant throughout the beam

11. Deflection of Beams
Appendix B

12. Finding the Spring Constant

13. Calculate spring constant

14. Vertical Translational Plates

15. Torsional Deflections
Pure Torsion; Every cross section of the bar is identical

16. Intrinsic Stress
Many thin film materials experience internal stress even when they are under room temperature and zero external loading conditions
In many cases related to MEMS structures, the intrinsic stress results from the temperature difference during deposition and use

17. Intrinsic Stress
The flatness of the membrane is guaranteed when the membrane material is under tensile stress

18. Intrinsic Stress
There are three strategies for minimizing undesirable intrinsic bending
Use materials that inherently have zero or very low intrinsic stress
For materials whose intrinsic stress depends on material processing parameters, fine tune the stress by calibrating and controlling deposition conditions
Use multiple-layered structures to compensate for stress-induced bending

19. Mechanical Variables of Concern
Force constant
flexibility of a given device
Mechanical resonant frequency
response speed of device
Hooke’s law applied to DC driving
Importance of resonant freq.
Limits the actuation speed
lower energy consumption at Fr

20. Types of Electrical-Mechanical Analysis
Given dimensions and materials of electrostatic structure, find
force constant of the suspension
structure displacement prior to pull-in
value of pull-in voltage
Given the range of desired applied voltage and the desired displacement, find
dimensions of a structure
layout of a structure
materials of a structure
Given the desired mechanical parameters including force constants and resonant frequency, find
dimensions
materials
layout design
quasistatic displacement

21. Analysis of Mechanical Force Constants
Concentrate on cantilever beam (micro spring boards)
Three types of most relevant boundary conditions
free: max. degrees of freedom
fixed: rotation and translation both restricted
guided: rotation restricted.
Beams with various combination of boundary conditions
fixed-free, one-end-fixed beam
fixed-fixed beam
fixed-guided beam
Fixed-free
Two fixed-
guided beams
Four fixed-guided beams

22. Examples

23. Boundary Conditions
Six degrees of freedom: three axis translation, three axis rotation
Fixed B.C.
no translation, no rotation
Free B.C.
capable of translation AND rotation
Guided B.C.
capable of translation BUT NOT rotation

24. A Clamped-Clamped Beam
Fixed-guided
Fixed-guided

25. A Clamped-Free Beam

26. One-end Supported, “Clamped-Free” Beams

27. Fixed-Free Beam by Sacrificial Etching
Right anchor is fixed because its rotation is completely restricted.
Left anchor is free because it can translate as well as rotate.
Consider the beam only moves in 2D plane (paper plane). No out-of-plane translation or rotation is encountered.

28. Force Constants for Fixed-Free Beams
Dimensions
length, width, thickness
unit in mm.
Materials
Young’s modulus, E
Unit in Pa, or N/m2.

29. Modulus of Elasticity Names Definition
Young’s modulus
Elastic modulus
Definition
Values of E for various materials can be found in notes, text books, MEMS clearing house, etc.

30. Large Displacement vs. Small Displacement
end displacement less than times the thickness.
Used somewhat loosely because of the difficulty to invoke large-deformation analysis.
Large deformation
needs finite element computer-aided simulation to solve precisely.
In limited cases exact analytical solutions can be found.

31. Force Constants for Fixed-Free Beams
Moment of inertia I (unit: m4)
I= for rectangular cross section
Maximum angular displacement
Maximum vertical displacement under F is
Therefore, the equivalent force constant is
Formula for 1st order resonant frequency
where is the beam weight per unit length.

32. Zig-Zag Beams Saves chip real-estate
Used to pack more “L” into a given footprint area on chip to reduce the spring constant without sacrificing large chip space.
Saves chip
real-estate

33. An Example

34. Order of Resonance
1st order: one node where the gradient of the beam shape is zero;
also called fundamental mode.
With lowest resonance frequency.
2nd order: 2 nodes where the gradient of the beam shape is zero;
3nd order: 3 nodes.
Frequency increases as the order number goes up.

35. Resonant frequency of typical spring-mass system
Self-mass or concentrated mass being m
The resonant frequency is
Check consistency of units.
High force constant (stiff spring) leads to high resonant frequency.
Low mass (low inertia) leads to high resonant frequency.
To satisfy both high K and high resonant frequency, m must be low.

36. Quality Factor
If the distance between two half-power points is df, and the resonance frequency if fr, then
Q=fr/df
Q=total energy stored in system/energy loss per unit cycle
Source of mechanical energy loss
crystal domain friction
direct coupling of energy to surroundings
distrubance and friction with surrounding air
example: squeezed film damping between two parallel plate capacitors
requirement for holes: (1) to reduce squeezed film damping; (2) facilitate sacrificial layer etching (to be discussed later in detail).
Source of electrical energy loss
resistance ohmic heating
electrical radiation