EE C245 – ME C218 Fall 2003 Lecture 13EE C245 - ME C218Introduction to MEMS DesignFall 2003Roger Howe and Thara SrinivasanLecture 13Alternative Transduction Principles2EE C245 – ME C218 Fall 2003 Lecture 13Today’s Lecture• Piezoelectric materials for MEMS: courtesy ofJustin Black (jblack@eecs) and Prof. R. M. White• Piezoresistive strain sensing in silicon:mechanism and device application• Thermal actuation: microtweezers• Reading:Senturia, S. D., Microsystem Design, KluwerAcademic Publishers, 2001, Chapter 18, pp. 470-477, Chapter 21, 570-578.3EE C245 – ME C218 Fall 2003 Lecture 13Origin of the Piezoelectric EffectSeveral views of an α-quartz crystalSiOX1X2X3ZYJ. Black and R. M. White4EE C245 – ME C218 Fall 2003 Lecture 13Origin of the Piezoelectric EffectSi atomO atoma / 4a / 2OPr >> aFor r >> a, the electric field at the point P is: The potential and electric field appear as if the charges are coincident at their center of gravity (point O)0434322=−≈+=−+rqrqEEEpπεπεEp≅ 0Introduction to Quartz Crystal Unit Design, Virgil Bottom, 1982.J. Black and R. M. White5EE C245 – ME C218 Fall 2003 Lecture 13Origin of the Piezoelectric Effect♦ Assume the applied force F causes the line OD to rotate counter clockwise by a small angle dθ♦ This strain shifts the center of gravity of the three positive and negative charges to the left and right, respectively♦ A dipole moment, p = qr, is created which has an arm (r) of:p = qr @ qa33/2dθFF60°dθP = NqrOD♦ Assuming the crystal contains N such molecules per unit volume, each subject to the same strain dθ, the polarization (or dipole moment per unit volume) is:P = Nqa33/2dθstrainpolarizationSi atomO atomJ. Black and R. M. White6EE C245 – ME C218 Fall 2003 Lecture 13Origins of the Piezoelectric Effect♦ For sufficiently small deformations, the polarization (P) is linearly related to the strain (S) by:P = gSwhere g is the piezoelectric voltage coefficient.♦ The polarization P equals the surface charge per unit area, or piezoelectric displacement.Converse Piezoelectric Effect♦ When a piezoelectric crystal is placed in an electric field, positive and negative ions are pushed in opposite directions and a dipole tends to rotate to align itself with the electric field.♦ The resulting motion gives rise to a strain S that is proportional to the electric field ES = dEwhere d is the piezoelectric charge coefficient.J. Black and R. M. White7EE C245 – ME C218 Fall 2003 Lecture 13Anisotropic Crystal Properties: Generalized Stress-Strain♦ In anisotropic materials a tensile stress can produce both axial and shear strain.♦ For example, a thin, X-cut rod of quartz subject to a tensile force will not only become longer and thinner, but it will also rotate about its longitudinal axis.♦ Since we have 6 components of stress (T) and 6 components of strain (S), 36 constants must be used to describe behavior in the general case.♦ Crystal symmetry (e.g. trigonal, hexagonal) greatly reduces the number of independent constants.Perspective and cross sectional views of α-quartzSiOJ. Black and R. M. White8EE C245 – ME C218 Fall 2003 Lecture 13X1X2X3ZYAnisotropic Crystal Properties: Generalized Stress-Strain⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡xyzxyzzzyyxxxyzxyzzzyyxxTTTTTTssssssssssssssssssssssssssssssssssssSSSSSS666564636261565554535251464544434241363534333231262524232221161514131211Quartz has threefold symmetry, physical properties repeat every 120°.Quartz is also symmetric about the X-axisConservation of energy requires sij= sji. Performing rotations based upon trigonal symmetry considerations, the compliance matrix reduces to 6 independent coefficients:For small deformations, stress (T) and strain (S)are related through the compliance matrix (s)⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡−−−=⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡xyzxyzzzyyxxxyzxyzzzyyxxTTTTTTsssssssssssssssssssSSSSSS)(22000020000000000000012111414444414143313131413112114131211J. Black and R. M. White9EE C245 – ME C218 Fall 2003 Lecture 13Anisotropic Crystal Properties: Piezoelectric ConstantsRecall that the strain (S) is related to the electric field (E) by the piezoelectric charge coefficient matrix (d)⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡zyxxyzxyzzzyyxxEEEddddddddddddddddddSSSSSS362616352515342414332313322212312111Applying the symmetry conditions for quartz, the piezoelectric strain matrix (d)simplifies to:⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡−−−=⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡zyxxyzxyzzzyyxxEEEdddddSSSSSS020000000000001114141111ZYXEXTENSIONALZYXSHEAR(about axis)ZYXFIELD (E)STRAIN (S)J. Black and R. M. White10EE C245 – ME C218 Fall 2003 Lecture 13Anisotropic Crystal Properties♦ Elastic modulus and compliance♦ Thermal conductivity♦ Electrical conductivity♦ Coefficient of thermal expansion♦ Dielectric constants♦ Piezoelectric constants♦ Optical index of refraction♦ Velocity of propagation of longitudinal waves♦ Velocity of propagation of shear wavesModes in QuartzJ. Black and R. M. White11EE C245 – ME C218 Fall 2003 Lecture 13Constitutive Equations for Piezoelectric MaterialsD = dTrT +εTES = sET + d Estrainstress electric fieldpiezoelectric strain coefficientscomplianceelectricdisplacementstresselectric fieldpiezoelectric strain coefficients (transpose)dielectricpermittivitySuperscripted material constants (e.g. sE) are those values obtained when the superscripted quantity is held constant.J. Black and R. M. White12EE C245 – ME C218 Fall 2003 Lecture 136,08046003948†24004112†4379†446011,300Velocity(m / s)5.607.552.651.887.644.645.853.26Density(kg / m3)0.1813-12.0 (d31)0.3P(VDF–TrFE)6.08.65.6 (d33)33.0Aluminum Nitride (AlN)7.566 - 730.11†4.7†17.2†39–46CouplingCoefficient K2(%)8.510-12 (d33)21.0Zinc Oxide (ZnO)1100-3200240-550 (d33)4.8 – 13.5PZT(PbZrTiO3)*4.52.3 (d11)10.7Quartz(SiO2)418.0 (d33)23.3Lithium Tantalate(LiTaO3)4419.2 (d33)24.5Lithium Niobate(LiNbO3)625-135082-145 (d33)11.0 - 27.5Barium Titanate (BaTiO3)*RelativePermittivityStrain
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