What is the Young’s Modulus of Silicon?M. HopcroftSpring 2006************************Silicon has a regular crystal structure, which is one of the reasons it is such anexcellent engineering material. It is an anisotropic crystal, so its properties aredifferent in different directions in the material relative to the crystal orientation.When designingmechanicalstructures insilicon, it isimportant tounderstand whichcrystal orientationsare relevant to thestructure and whatcorrespondingvalues formechanicalproperties shouldbe used forcalculations.M. Hopcroft 2006What is the Crystal Orientation in a Silicon Wafer?The crystal orientation of a wafer is defined by the plane of its top surface.(100) wafers are most common, but other orientations are available.the top surface ofthis wafer is the(100) crystal planethe directionnormal to thetop surface isthe [100]directiona (100) Si wafery[011][001]45°x[011]A typical (100) wafer with in-planedirections indicatedHopcroft 2006N. Maluf, An Introduction to Microelectromechanical SystemsEngineering, 1st ed. Boston: Artech House, 2000.M. Hopcroft 2006Fine. So What Value Should I Use for E of Silicon?• Silicon is an anisotropic crystal, so its Young’s modulus varies in differentdirections in the material relative to the crystal orientation.• Silicon has cubic symmetry, so the 3D direction-dependent properties can bedescribed with a 6x6 matrix with only 3 independent constants (eitherstiffnesses Cij or compliances Sij). Because silicon is such an importanteconomic material, these values have been investigated thoroughly. The bestvalues of these constants are:C: 109 PaS: 10-12 PaC11C12C44S11S12S44Si165.763.979.67.68-2.1412.6W. A. Brantley, \"Calculated elastic constants for stress problems associated with semiconductor devices,\"Journal of Applied Physics, vol. 44, pp. 534-535, 1973.J. J. Wortman and R. A. Evans, \"Youngs' Modulus, Shear Modulus and Poisson's Ratio in Silicon andGermanium,\" Journal of Applied Physics, vol. 36, pp. 153-156, 1965.H. J. McSkimin and J. P. Andreatch, \"Elastic Moduli of Silicon vs Hydrostatic Pressure at 25.0 !C and - 195.8 !C,\" Journal of Applied Physics, vol. 35, pp. 2161-2165, 19.W. P. Mason, Physical acoustics and the properties of solids. Princeton, NJ USA: Van Nostrand, 1958.• For FEM calculations, use the stiffness matrix. Be sure to type it in yourself!Many common programs (ANSYS, FEMLAB) do not have default values.M. Hopcroft 2006For design calculations:• Axial tension/compression: Use the E value for the direction oftension/compression.l=cos(\")*m=cos(#)n=cos($)& 11) 222222=S11%2((S%S%Slm+mn+ln))(' 1112244+* E\"#$1=S11E100 11# 1& 12# 1& =S11\"%(S11\"S12)\"S44( =S11\"%(S11\"S12)\"S44( $ ' $ E11022E1113 2' * \"direction cosines\": cosine of the angle between the direction of interest and x,y,z axes (the <100> directions)J. F. Nye, Physical properties of crystals : their representation by tensors and matrices. Oxford: Oxford University Press, 1985.• For example, in a (100) wafer:• for “x or y axis” (parallel to flat), use E110 = 169 GPa• for “off-axis” (45° diagonal to flat), use E100 = 130 GPa•Small deflection of a long, thin beam: This is basically axial tension andcompression, so use the E value for the direction of the neutral axis.• Small deflection of a plate edge: Use a “plate modulus” for the major axis ofE\"#$bending: , with Poisson’s ratio of ! = 0.28 for (100). [Brantley 1973]1%&M. Hopcroft 2006 • Thin film stress/strain calculations for a thin film membrane or bendingsubstrate (incl. Stoney’s equation calculations): Use the symmetric Biaxialmodulus Bijk for the appropriate symmetric crystal plane, either (100) or (111)(the (110) plane is not symmetric).2C122B100=C11+C12\"C11B111=6C44(C11+2C12)C11+2C12+4C44W. D. Nix, Stanford MSE 353 Material Properties of Thin FilmsFor example, B100 = 179.4 GPa• For stress concentration calculations or multi-directional polycrystalline situations (unusual), use a Voigt or Reuss volume average.• For hydrostatic loads, use a Bulk modulus, B = 97.83 GPa.J. Diz and M. Humbert, \"Practical aspects of calculating the elastic properties of polycrystals from the texture according to different models,\"Journal of Applied Crystallography, vol. 25, pp. 756-760, 1992.George, \"Elastic constants and moduli of diamond cubic Si,\" in Properties of Crystalline Silicon, vol. 20, EMIS Datareviews, R. Hull, Ed. London:INSPEC, IEE, 1997, pp. 98.• Most silicon wafers are not pure silicon. Electronic doping changes the elasticbehaviour in predictable ways. Typically the changes are negligible (1-5%).See J. J. Hall, \"Electronic effects in the constants of n-type silicon,\" Physical Review, vol. 161, pp. 756, 1967, and references cited therein.For more information, see:•H. J. McSkimin, \"Measurement of Elastic Constants at Low Temperatures by Means of Ultrasonic Waves--Data for Silicon and Germanium SingleCrystals, and for Fused Silica,\" Journal of Applied Physics, vol. 24, pp. 988-997, 1953.•H. J. McSkimin, W. L. Bond, E. Buehler, and G. K. Teal, \"Measurement of the Elastic Constants of Silicon Single Crystals and Their ThermalCoefficients,\" Physical Review, vol. 83, pp. 1080, 1951.•C. Bourgeois, E. Steinsland, N. Blanc, and N. F. de Rooij, \"Design of resonators for the determination of the temperature coefficients of elasticconstants of monocrystalline silicon,\" Proceedings of the 1997 IEEE International Frequency Control Symposium, 1997.•Stanford MSE 353 (Nix / Barnett)M. Hopcroft 2006•Stanford MSE 208 (Dauskart)
