Report

Solid mechanics Define the terms Calculate stresses in deposited thin films using the disk Stress method Deformation Strain Thermal strain Thermal expansion coefficient Appropriately relate various types of stress to the correct corresponding strain using elastic theory Give qualitative descriptions of how intrinsic stress can form within thin films Calculate biaxial stress resulting from thermal mismatch in the deposition of thin films Why? Solid mechanics... Why? Why? Why? Why is this thing bent? Thermal actuator produced by Southwest Research Institute And these? A bi-layer of TiNi and SiO2. (From Wang, 2004) Why? Membrane is piezoresistive; i.e., the electrical resistance changes with deformation. Adapted from MEMS: A Practical Guide to Design, Analysis, and Applications, Ed. Jan G. Korvink and Oliver Paul, Springer, 2006 A simple piezoelectric actuator design: An applied voltage causes stress in the piezoelectric thin film stress causing the membrane to bend Why? Hot arm actuator + e - Joule heating leads to different rates of thermal expansion, in turn causing stress and deflection. Zap it with a voltage here… i + e - How much does it move here? ω Stress and strain A = w·t t P P w w δ L P P σ=—=— A [A ] ε=— Normal stress L wt Typical units Dimensions [F ] δ [F ] [L ]2 N m 2 Pa Normal strain Dimensions [L ] (dimensionless) [L ] Typical units μ-strain = 10-6 Elasticity How are stress and strain related to each other? P X fracture F = kx σ plastic (permanent) deformation E σ= E ε L elastic (permanent) deformation brittle ductile E δ P X fracture ε Young’s modulus (Modulus of elasticity, Elastic modulus) Elasticity Strain in one direction causes strain in other directions y x εy = -ν εx Poisson’s ratio Stress generalized Stress is a surface phenomenon. z σz ΣF = 0, ΣMo = 0 τzy τzx τxz σy τxy σx x τxy = τyx τyz = τzy τzx = τxz τyz τyx y σ : normal stress Force is normal to surface σx stress normal to x-surface τ : shear stress Force is parallel to surface τxy stress on x-surface in y-direction Strain generalized Essentially, strain is just differential deformation. Δy Δy + dΔy Deforms Δx Δx + dΔx Break into two pieces: ux = dux ux + dx + dy θ2 θ1 dx uniaxial strain duy Shear strain is strain with no volume change. shear strain u: displacement du x x dx xy du x du y 1 2 dx dy Relation of shear stress to shear strain Just as normal stress causes uniaxial (normal) strain, shear stress causes shear strain. τxy = G γxy dux τyx τxy τxy θ2 τyx duy θ1 shear modulus G E 2(1 ) • Si sabes cualquiera dos de E, G, y ν, sabes el tercer. • Limits on ν: Magic Algebra Box 0 < ν < 0.5 ν = 0.5 incompressible Generalized stress-strain relations The previous stress/strain relations hold for either pure uniaxial stress or pure shear stress. Most real deformations, however, are complicated combinations of both, and these relations do not hold εx = [ x E x normal strain due to x normal stress ] + [ -ν y E ] + [ -ν x normal strain due to y normal stress Deforms z E ] x normal strain due to z normal stress τxy = G γxy Generalized Hooke’s Law For a general 3-D deformation of an isotropic material, then 1 yz γyz = G 1 zx γzx = G 1 x y z εx = E 1 y z x εy = E 1 z x y εz = E γxy = Generalized Hooke’s Law 1 xy G Special cases • Uniaxial stress/strain σ = Eε • No shear stress, todos esfuerzos normales son iguales volume strain σx = σy = σz = σ = K•(ΔV/V) bulk modulus • Biaxial stress Stress in a plane, los dos esfuerzos normales son iguales σx = σy = σ = [E / (1 - ν)] • ε biaxial modulus Elasticity for a crystalline silicon The previous equations are for isotropic materials. Is crystalline silicon isotropic? E Cij Compliance coefficients xx C11 C12 C11 21 22 yy C12 zz C12 C12 31 32 xyxy C041 C042 xzxz C0 C0 52 51 yzyz C061 C062 C12 13 C12 23 C11 33 C043 C053 C063 0x15 C014 C C024 C0y25 0z35 C034 C C44 C0xy45 C054 Cxz55 44 C064 C0yz65 C016 x C026 y C036 z C046 xy C056 xz C66 44 yz For crystalline silicon C11 = 166 GPa, C12 = 64 GPa and C44 = 80 GPa Te toca a ti Assuming that elastic theory holds, choose the appropriate modulus and/or stress-strain relationship for each of the following situations. 1. A monkey is hanging on a rope, causing it to stretch. How do you model the deformation/stress-strain in the rope? Uniaxial stress/strain 2. A water balloon is being filled with water. How do you model the deformation/stress-strain in the balloon membrane? Biaxial stress/strain 3. A nail is hammered into a piece of plywood. How do you model the deformation/stress-strain in the nail? Uniaxial stress/strain 4. A microparticle is suspended in a liquid for use in a microfluidic application, causing it to compress slightly. How do you model the deformation/stress-strain in the microparticle? Use bulk modulus (no shear, all three normal stresses the same) 5. A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the thin film? Biaxial stress/strain 6. A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the wafer? Anisotropic stress/strain (Using Cij compliance coefficients) 7. A thin film is deposited on a much thicker glass substrate. How do you model the deformation/stress-strain in the glass substrate? Generalized Hooke’s Law. I.e., ε = (1/E)(σx – ν(σy + σz)) etc. Thermal strain Thermal Expansion Most things expand upon heating, and shrink upon cooling. δ(T) = αT (T-T0) Notes: d T T dT Thermal expansion coefficient • αT ≈ constant ≠ f(T) If no initial strain ε(T) ≈ ε(T0) + αT (T-T0) • Thermal strain tends to be the same in all directions even when material is otherwise anisotropic. Solid mechanics of thin films Adhesion Ways to help ensure adhesion of deposited thin films: • Ensure cleanliness • Increase surface roughness • Include an oxide-forming element in between a metal deposited on oxide Stress in thin films positive (+) Negative (-) Tension Compression Tension headache Stress in thin films Two types of stress Intrinsic stress Extrinsic stress Also known as growth stresses, these develop during as the film is being formed. These stresses result from externally imposed factors. Thermal stress is a good example. Doping Sputtering Microvoids Gas entrapment Polymer shrinkage Thermal stress in thin films Consider a thin film deposited on a substrate at a deposition temperature, Td. (Both the film and the substrate are initially at Td.) thin film deposited at Td Initially the film is in a stress free state. The film and substrate are then allowed to cool to room temperature, Tr Since the two materials are hooked together, they both experience strain as they cool. the same ____________ substrate both cooled to Tr εboth = εsubstrate or εfilm ? εsubstrate = αT,s(Tr - Td) = εfilm = αT,f (Tr - Td) + εmismmatch εmismmatch = αT,s(Tr - Td) - αT,f (Tr - Td) = (αT,f - αT,s)(Td - Tr) Thermal stress in thin films How would you relate σmismatch to εmismatch? Biaxial stress/strain σmismatch = [E / (1-ν)]·εmismatch = [E / (1-ν)]·(αT,f - αT,s)(Td - Tr) tension If αT,f > αT,s σmismatch = (+) or (-) Film is in ___________________. compression If αT,f < αT,s σmismatch = (+) or (-) Film is in ___________________. Thin film Initially stress free cantilever Sacrificial layer σmismatch > 0 σmismatch < 0 Stress in thin films Compression or tension? Compression or tension? (a) (b) (a) Stress in SiO2/Al cantilevers (b) Stress in SiO2/Ti cantilevers [From Fang and Lo, (2000)] αT,Al >, <, = αT,SiO2 ? αT,Ti >, <, = αT,SiO2 ? Stress in thin films How were these fabricated? (a) (b) (a) Stress in SiO2/Al cantilevers (b) Stress in SiO2/Ti cantilevers [From Fang and Lo, (2000)] Te toca a ti Show that the biaxial modulus is given by E/(1 – ν) Pistas: • Remember what the assumptions for “biaxial” are. • In thin films you can always find one set of x-y axes for which there is only σ and no τ. Measuring thin film stress The disk method Stressed wafer (after thin film) Unstressed wafer (before thin film) R radius of curvature , R = _________________________ wafer thickness T = _________________________ and thin film thickness t = _________________________. Assumptions: The film thickness is uniform and small compared to the wafer thickness. The stress in the thin film is biaxial and uniform across it’s thickness. Ths stress in the wafer is equi-biaxial (biaxial at any location in the thickness). The wafer is unbowed before the addition of the thin film. Wafer properties are isotropic in the direction normal to the film. The wafer isn’t rigidly attached to anything when the deflection measurement is made. 2 E T 1 6Rt Biaxial modulus of the wafer strain at wafer/film interface