EE C245 – ME C218Introduction to MEMS DesignFall 2007Fall 2007Prof Clark TC NguyenProf. Clark T.-C. NguyenDept of Electrical Engineering & Computer SciencesDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyBerkeley, CA 94720yLt 11 Mhil P tiEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 1Lecture 11: Mechanical PropertiesLecture Outline• Reading: Senturia, Chpt. 8gp• Lecture Topics:ª Stress, strain, etc., for isotropic materialsªThin films: thermal stress residual stress and stress ªThin films: thermal stress, residual stress, and stress gradientsª Internal dissipationpª MEMS material properties and performance metricsEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 2Vertical Stress Gradients• Variation of residual stress in the direction of film growth• Can warp released structures in z-directionEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 3ElasticityElasticityEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 4Normal Stress (1D)If the force acts normal to a surface then the surface, then the stress is called a normal stressnormal stresszσzσyz∆zxyDiff ti l σx∆x∆y∆zEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 5Differential volume elementStrain (1D)EE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 6The Poisson RatioEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 7Shear Stress & Strain (1D)EE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 82D and 3D Considerations• Important assumption: the differential volume element lb is in static equilibrium → no net forces or torques (i.e., rotational movements)rotational movements)ª Every σ must have an equal σ in the opposite di ti th th direction on the other side of the elementªFor no net torque, the ªq,shear forces on different faces must also be matched as Stresses acting on a differential volume elementalso be matched as follows:τ= ττ= ττ= τdifferential volume elementEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 9τxy= τyxτxz= τzxτyz= τxy2D Strain• In general, motion consists ofª rigid-body displacement (motion of the center of mass)ªrigidbody rotation (rotation about the center of mass)ªrigid-body rotation (rotation about the center of mass)ª Deformation relative to displacement and rotationArea element experiences both di l d displacement and deformation•M k ih dil •Must work with displacement vectors• Differential definitionf xi l str in:()()uxuxxuxxx∂−Δ+εEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 10of axial strain:()()xxxxxx∂=Δ=ε2D Shear StrainEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 11Ex: Volume Change for Uniaxial StressStresses acting on a differential differential volume elementEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 12Isotropic Elasticity in 3D• Isotropic = same in all directions• The complete stress-strain relations for an isotropic elastic ppsolid in 3D: (i.e., a generalized Hooke’s Law)()[]11()[]zyxxEσσνσε+−=11xyxyGτγ1=1()[]xzyyEσσνσε+−=1yzyzGτγ1=()[]yxzzEσσνσε+−=1zxzxGτγ1=()[]yEGBasically, add in off-axis strains from l i h di iEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 13normal stresses in other directionsImportant Case: Plane Stress• Common case: very thin film coating a thin, relatively rigid substrate (e.g., a silicon wafer)•At regions more than 3 thicknesses from edges the top At regions more than 3 thicknesses from edges, the top surface is stress-free →σz= 0• Get two components of in-plane stress:pfp)]0()[1( +−=yxxEσνσεEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 14)]0()[1( +−=xyyEσνσεImportant Case: Plane Stress (cont.)• Symmetry in the xy-plane →σx= σy= σ• Thus, the in-plane strain components are: εx= εy= εppywhereEσσ])[1(EEEx′=−=−=ννσσε)]1([])[1(and where′EEBiaxial Modulus =∆ν−=1EBiaxial Modulus =EE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 15Edge Region of a Tensile (σ>0) FilmNet non-zero in-plane force (that ld)At free edge, in-plane force Film must be bent we just analyzed)must be zeroThere’s no Poisson back, hereThere s no Poisson contraction, so the film is slightly thicker herethicker, hereDiscontinuity of stress at the attached corner iPeel forces that can peel the film EE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 16→ stress concentrationpoff the surfaceLinear Thermal Expansion• As temperature increases, most solids expand in volume• Definition: linear thermal expansion coefficientdxTεα=[Kelvin-1]Linear thermal expansion coefficient=∆Remarks:dTT[]expansion coefficient• αTvalues tend to be in the 10-6to 10-7range• Can capture the 10-6by using dimensions of μstrain/K, where 10-6K-1= 1 μstrain/Kwhere 106K1= 1 μstrain/K• In 3D, get volume thermalexpansion coefficientTVTΔ=Δα3expans on coeff c ent• For moderate temperature excursions, aTcan be treated as t t f th t i l b t i t lit it i f ti TVTΔα3EE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 17a constant of the material, but in actuality, it is a function of temperatureαTAs a Function of Temperature[Madou, Fundamentals of Microfabrication, CRC Press, 1998]EE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 18• Cubic symmetry implies that α is independent of directionThin-Film Thermal StressThin Film (αTf)Silicon Substrate (αTs= 2.8 x 10-6K-1)Substrate much thicker than thin film• Assume film is deposited stress-free at a temperature Td, h h hl h ld then the whole thing is cooled to room temperature Tr• Substrate much thicker than thin film → substrate dictates the amount of contraction for both it and the thin filmthe amount of contraction for both it and the thin filmEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 19Linear Thermal ExpansionEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 20MEMS Material PropertiesMEMS Material PropertiesEE C245: Introduction to MEMS Design Lecture 11 C. Nguyen 10/2/08 21Material Properties for MEMS√(E/ρ) is custic acoustic velocityEE C245: Introduction to MEMS
View Full Document
Unlocking...