Mechanics of Micro Structures Micro Actuators, Sensors, Systems

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

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)

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

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

General Relation Between Tensile Stress and Strain

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

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

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

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

Deflection of Beams Appendix B

Finding the Spring Constant

Calculate spring constant

Vertical Translational Plates

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

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

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

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

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

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

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

Examples

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

A Clamped-Clamped Beam Fixed-guided Fixed-guided

A Clamped-Free Beam

One-end Supported, “Clamped-Free” Beams

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.

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

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.

Large Displacement vs. Small Displacement end displacement less than 10-20 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.

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.

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

An Example

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.

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.

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