MEMS和微系统设计和微系统设计课程内容课程内容n nMEMS概述及MEMS设计的概述n n工艺简要回顾n n系统设计、工艺设计及版图设计n n主要的机械、电子元件及其设计基础n n多域耦合设计：以机电耦合为例子n n器件性能的估计n n简单的其他域的元件及其简要设计要点n n设计实例 第第4讲主要内容讲主要内容 (3)1、弹簧设计原理及计算例子2、薄膜设计原理及计算例子3、电容设计原理及计算例子4、电阻设计原理及计算例子5、压电模型n电容变化n静电力图2-17电容式微传感器的基本结构 平行板电容器的电容为电容敏感原理 式中 A为极板面积 为真空介电常数 为极板间介质的相对介电常数 当介质为空气时， ； 为两极板间距离间隙变化型：改变两极板间隙面积变化型：改变形成电容的有效面积A介质变化型：改变两极间介质的介电常数 间隙变化型电容式微传感器利用泰勒级数展开，由麦克劳林公式可得 略除高阶无穷小项，得这时传感器的灵敏度和非线性误差分别为：采用差动电容结构可以大大减小传感器输出的非线性：(2-12)(2-13)(2-14)(2-15) 在小位移情况下，外加作用和成比例关系，可见电容的倒数差及电容的差除和都与输入作用力成线性关系。 式(2-14)表明，用电容的差除和表达传感器的性能，其输出还要受到介质介电常数的影响。 式(2-15)表明电容差除和只受电容极板间隙和间隙变化的影响。目前，硅电容变送器普遍采取式(2-15)的方法来描述传感器的性能。 其他的电容变化形式其他的电容变化形式n n变面积电容器 A example: calculate to C and the shift of C两种电容变化形式的变化量对比两种电容变化形式的变化量对比 （电容原值、导线的电容值、电容变化值）（电容原值、导线的电容值、电容变化值）Wire：L=1m, r=0.2mm, d=1mmgap=g=1Thickness=t=2finger length=L=100overlap length x=75电容电容readout位置检测和速度检测位置检测和速度检测Why modulate v(t)?Ideal buffer: cin=0Matched Air-Gap Reference Capacitors Simple Capacitor Divider (con.)matched air-gap reference capacitoroffsetsignalCapacitor Divider With Differential ExcitationWhy modulate v+ and v- ?Ideal buffer: cin=0Impedance divider with superposition:Improved Capacitive Divider (cont.)no offset!distortionThe capacitive Half -BridgeImpedance divider with superposition:The capacitive Half Bridge (cont.)Simplify expression: No offset， 2x signal increaseParasitic CapacitancesSurface Surface micromachinedmicromachined z-axis parallel-plate capacitor z-axis parallel-plate capacitorEquivalent circuitC Cpppp (x): nominal | plate sense capacitor(x): nominal | plate sense capacitorC Cf1 f1 (x): fringe capacitance (varies with plate displacement)(x): fringe capacitance (varies with plate displacement)C Cf2 f2 :fringe capacitance between upper plate (connected to anchor plane) :fringe capacitance between upper plate (connected to anchor plane) and lower plate slight dependence on xand lower plate slight dependence on xC Cpupu :parasitic capacitance from upper plate to substrate :parasitic capacitance from upper plate to substrate C Cpl pl : parasitic capacitance from lower plate to substrate : parasitic capacitance from lower plate to substrate Velocity SensingFundamental current-voltage relationship for a Fundamental current-voltage relationship for a time-varying capacitor:time-varying capacitor:Consider special case: v=Consider special case: v=v vp p =constant =constantused in high-quality capacitance microphonesused in high-quality capacitance microphonesVelocity Sensing (cont.)Sense capacitors time variation:Sense capacitors time variation:Parallel-plate sense capacitor with gap gParallel-plate sense capacitor with gap go o : :Harmonic motion:Harmonic motion:Some NumbersSurface Surface micromachinedmicromachined capacitor: capacitor: Is this real?noise in buffer ampWorld Record CapacitivePosition-Sense Resolution*Analog Devices ADRS-150 vibratory rate gyroscope John Geen ,Steve Sherman, John Chang, and Steve Lewis, IEEE J. Solid-State Circuits, 37, Dec. 2002, 1860-1866Full scale Corillis-induced displacement=20Sense capacitance 1000fFMinimum detectable capacitance change 12 zF =0.012 aFNominal sense gap = 1.6 m Minimum displacement: 16 fm ! * Surface micromachining class audio frequency band EE C245-ME C218 Fall 2003 Lecture 12Is ADL Splitting Electrons?A At t V V+ + =5V, the charge on the sense capacitor is: =5V, the charge on the sense capacitor is: q qs s =c=c+ + v v+ + =(1000fF)(5V)=5000fC =(1000fF)(5V)=5000fCNumber of electrons at Number of electrons at Minimum detectable change in sense charge:Minimum detectable change in sense charge:Minimum detected change in number of electronsMinimum detected change in number of electrons: :n电容变化n静电力变间隙电容驱动器的基本理论变间隙电容驱动器的基本理论变间隙电容驱动器的基本理论变间隙电容驱动器的基本理论Basic physics of Electrostatic ActuationBasic physics of Electrostatic Actuation Two ways to change the energy:Two ways to change the energy: 1.Change the charge q 1.Change the charge q 2.change the separation x 2.change the separation xNote: we assume that the plates are supported elastically ,so they Note: we assume that the plates are supported elastically ,so they dont collapse .dont collapse . Charge-Control Case (cont.)Stored energy:Force (attractive ,internal):Voltage:Independent of the gap!constantElectrostatic Force (Voltage Control)Find co-energy in terms of voltageVariation of co-energy with respect to gap yields v.s. force:Variation of co-energy with respect to voltage yields chargeas expectedLinearizing the Voltage Square-Lawn nPolarize the capacitor by applying a DC offset voltage VP together Polarize the capacitor by applying a DC offset voltage VP together with a (small) signal voltage with a (small) signal voltage V Vsigsig (t) VP (t) VPDC offsetneglect(small)The Differential Electrostatic ActuatorNet force on suspended center electrode is the differenceParallel Plate Capacitive Nonlinearityn nExample: laterally driven spring suspended plate (eventually with Example: laterally driven spring suspended plate (eventually with balanced electrodes )balanced electrodes )n nNomenclatureNomenclatureConductives t r u c t u r eelectrodeValueAC or signal component (lower case variable subscript)DC Component (upper case variable: upper case subscript)Parallel Plate Capacitive Nonlinearityn nExample: clamped-clamped laterally driven beam Example: clamped-clamped laterally driven beam with balanced electrodeswith balanced electrodes Expression for Expression for Expand the Taylor Series further ConductivestructureelectrodeParallel plate Capacitive NonlinearityParallel Plate Capacitive Nonlinearityn nRetaining only terms at the drive frequency:Retaining only terms at the drive frequency:n nThese two together mean that this force acts These two together mean that this force acts against the spring restoring force!against the spring restoring force! A negative spring constant A negative spring constant since it derives from V since it derives from VP P we call it the electrical we call it the electrical stiffness, given bystiffness, given by:Drive force arising from the input excitation voltage at the frequency of this voltageProportional to displacement900 phase-shifted from drive, so in phase with displacementElectrical stiffness ,Ke eThe electrical stiffness ke behaves like any other stiffnessIt affects resonance frequency:Frequency is now a function of dc-bias Vp1Can One Cancel Ke with Two Electrodes?n nWhat if we dont like the dependence of What if we dont like the dependence of frequency on Vfrequency on VP P ? ?n nCan we cancel KC via a differential input Can we cancel KC via a differential input electrode configuration ?electrode configuration ?n nIf we do a similar analysis for FIf we do a similar analysis for Fd2d2 at Electrode at Electrode 2:2: Subtracts from the Fd1 term ,as expectedAdd to the quadrature term Kcs add, no matter the electrode configuration!The capacitive Half -BridgeImpedance divider with superposition:The capacitive Half Bridge (cont.)Simplify expression:Electrostatic force:Electrostatic Force (Cont.)Output voltage is proportional to the displacement (for x0Parallel plate Ke 0第第4讲主要内容讲主要内容 (3)1、弹簧设计原理及计算例子2、薄膜设计原理及计算例子3、电容设计原理及计算例子4、电阻设计原理及计算例子5、压电模型 1、金属的电阻改变：由材料几何尺寸的变化引起的；与 相关 2、半导体的电阻改变：由材料受力后电阻率的变化引起，与 相关； 3、半导体的灵敏度因子比金属的高得多，一般在70-170之间当电阻为立体结构时，有当电阻为立体结构时，有立体单元电阻的应力图(7-6)其中R= 代表与应力分量= （如图7.13）相对应的一个无限小的立方压电电阻晶体单元的电阻变化。 式7-6、7-7 立体电阻的压阻系数(7-7)得出： 若电阻为薄膜电阻，在正交坐标系中，当坐标轴与晶轴一致时，电阻的相对变化与应力的关系为 表示纵向应力 为横向应力 表示 、 垂直方向上的应力，它比 和 小很多，一般都略去。 、 、 分别为 、 、 相对应的压阻系数， 为纵向压阻系数， 为横向压阻系数。 当电阻处于任意晶向P时，如果有纵向应力 沿此方向作用在单晶硅电阻上，则会引起纵向压阻系数 ，如果电阻上同时作用有和电阻方向垂直的横向应力 ，则会引起横向压阻系数 ，那么任意晶向的压阻系数为 （2-6）（2-7）式中， 、 、 分别为单晶硅晶轴上的纵向压阻系数、横向压阻系数和剪切压阻系数；、 、 分别为电阻的纵向应力相对于晶体主轴坐标系中的方向余弦； 、 、 分别为电阻的横向应力相对于晶体主轴系中的方向余弦 。n nRelative resistance change can be expressed by the longitudinal Relative resistance change can be expressed by the longitudinal and transverse and transverse piezoresistivepiezoresistive coefficients coefficientsn nPiezoresistorsPiezoresistors are often aligned to the wafer flat of (100) wafers, are often aligned to the wafer flat of (100) wafers, which is in the 110 direction. Senturia,p.473 provides the result which is in the 110 direction. Senturia,p.473 provides the result of coordinate transformations: of coordinate transformations: Silicon piezoresistive coefficientsn nFunction of type ,doping, and temperatureFunction of type ,doping, and temperaturen nLongitudinal and transverse coefficients in110 directionLongitudinal and transverse coefficients in110 directionn-type 11.7 -102.2 53.4 -13.6P-type 7.8 6.6 -1.1 138.1Units -cm, 10-1Pa-1 values are at T=25 0Cn-type P-type 一般地，当晶面为一般地，当晶面为(100)(100)时，有时，有表7-9 P型压电阻在各方向的压阻系数晶面取向取向LT(100)+0.6644-0.3344(100)+0.5440(100)+0.544-0.544(100)+0.02440.0244Piezoresistor Placementn nBulk micromachined diaphragm pressure sensor电阻变化的电阻变化的read-out公式?n n举例计算电阻的变化导致电压的变化第第4讲主要内容讲主要内容 (3)1、弹簧设计原理及计算例子2、薄膜设计原理及计算例子3、电容设计原理及计算例子4、电阻设计原理及计算例子5、压电模型Origin of Piezoelectric Effectn nSeveral views of anSeveral views of an -quartz crystal -quartz crystalOrigin of Piezoelectric Effectn nFor ra, the electric field at the point P is:For ra, the electric field at the point P is:n nThe potential and electric field appear as if the charges The potential and electric field appear as if the charges are coincident at their center of gravity (point O)are coincident at their center of gravity (point O)Origin of Piezoelectric Effectn nAssume the applied force F causes Assume the applied force F causes the line OD to rotate counter the line OD to rotate counter clockwise by a small angle clockwise by a small angle n nThis strain shifts the center of This strain shifts the center of gravity of the three positive and gravity of the three positive and negative charges to the left and negative charges to the left and right, respectivelyright, respectivelyn nA dipole moment, p=A dipole moment, p=qrqr, is created , is created which has an arm (r) of:which has an arm (r) of: p= p=qrqr qa3 qa33/23/2n nAssuming the crystal contains N Assuming the crystal contains N such molecules per unit volume, such molecules per unit volume, each subject to the same strain each subject to the same strain , the polarization (or dipole , the polarization (or dipole moment per unit volume) is:moment per unit volume) is: polarizationstrainOrigin of Piezoelectric Effectn nFor sufficiently small deformations , polarization (For sufficiently small deformations , polarization (p p) is ) is linearly related to the strain (linearly related to the strain (s s) by: ) by: p p= =g gs s where g is the piezoelectric voltage coefficient. where g is the piezoelectric voltage coefficient.Converse Piezoelectric EffectConverse Piezoelectric Effectn nWhen a piezoelectric crystal is placed in an electric field, When a piezoelectric crystal is placed in an electric field, positive and negative ions are pushed in opposite positive and negative ions are pushed in opposite directions and a dipole tends to rotate to align itself with directions and a dipole tends to rotate to align itself with the electric field.the electric field.n nThe resulting motion gives rise to strain The resulting motion gives rise to strain s s that is that is proportional to electric field Eproportional to electric field E S S= =dEdE where d is the piezoelectric charge coefficient. where d is the piezoelectric charge coefficient.Anisotropic Crystal Properties: Generalized Stress-Strainn nInIn anisotropic materials a tensile anisotropic materials a tensile stress can produce both axial and stress can produce both axial and shear strain.shear strain.n nFor example, a thin, x- cut rod of For example, a thin, x- cut rod of quartz subject to a tensile force will quartz subject to a tensile force will not only become longer and thinner, not only become longer and thinner, longitudinal axis.longitudinal axis.n nSince we have 6 components of Since we have 6 components of stress (T) and 6 components of stress (T) and 6 components of strain (S), 36constants must be used strain (S), 36constants must be used to describe behavior in the general to describe behavior in the general case.case.n nCrystal symmetry (e.g. Crystal symmetry (e.g. trigonaltrigonal, , hexagonal) greatly reduces the hexagonal) greatly reduces the number of independent constants.number of independent constants.Anisotropic Crystal Properties: Generalized Stress-Strainn nFor small deformations, stress (T) and strain (S) For small deformations, stress (T) and strain (S) are related though the compliance matrix (s)are related though the compliance matrix (s)n nConservation of energy requires Conservation of energy requires s sij ij= =s sji ji. Performing . Performing rotations based upon rotations based upon trigonaltrigonal symmetry considerations, symmetry considerations, the compliance matrix reduces to 6 independent the compliance matrix reduces to 6 independent coefficients: coefficients: Quartz has threefold symmetry, physical properties repeat every 1200.Quartz is also symmetric about the x-axis Anisotropic Crystal Properties: Generalized Stress-Strainn nRecall thatRecall that the strain (S) is related to the electricthe strain (S) is related to the electric (E) by the (E) by the piezoelectric charge coefficient matrix (d)piezoelectric charge coefficient matrix (d)n nApplying the symmetry conditions for Applying the symmetry conditions for quartz, the piezoelectric strain matrix quartz, the piezoelectric strain matrix (d) simplifies to:(d) simplifies to:Anistropic Crystal Properties Elastic modulus and compliance Thermal conductivity Electrical conductivity Coefficient of thermal expansion Dielectric constants Piezoelectric contants Optical index of refraction Velocity of propagation of shear wavesConstitutive Equations for Piezoelectric Materials Superscripted material constants (e.g.sE) are those values obtained when superscripted quantity is held constant.Piezoelectric strainCoefficients (transpose)DielectricpermittivityProperties Common PiezoelectricsSAW Devices The stress-free boundary condition imposed by the surface of a crystal gives rise to an acoustic mode known as a surface acoustic wave (Rayleigh wave) SAW energy is confined to within one wavelength of the surface The components of surface particle motion, Ux and Uz, are 90 out of phase, and Uz Ux SAWs can be excited by interdigitated transducers (IDTs) patterned on the surface of piezoelectric crystals* IDT geometry allows construction of delay lines, convolvers, correlators, pluse compressors, filtersUsed by the billions in communications and electronic system; invented by Prof. R. M. White, EECS Dept UC Brekeley, 1965Derivation of Equivalent BVD CircuitnAssume piezoid is a thin plate thickness d with infinite extent in x and z(1-d solution) The equation of motion for the particle displacement is:Equivalent BVD Circuit Impedance governed by transducer area Typical SMR values are Co=1.91pF, Ca=0.80pF, Ra=1.14 ,and La=123.6nH. These values correspond to a Q of approximately 1000+