MEMS技术 第四讲 电子零件原理(2).ppt

1、MEMS和微系统设计,课程内容,MEMS概述及MEMS设计的概述 工艺简要回顾 系统设计、工艺设计及版图设计 主要的机械、电子元件及其设计基础 多域耦合设计：以机电耦合为例子 器件性能的估计 简单的其他域的元件及其简要设计要点 设计实例,第4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,电容变化 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

2、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(2-1

3、5)的方法来描述传感器的性能。,其他的电容变化形式,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 （电容原值、导线的电容值、电容变化值）,Wire：L=1m, r=0.2mm, d=1mm,gap=g=1 Thickness=t=2 finger length=L=100 overlap length x=75,电容readout位置检测和速度检测,Why modulate v(t)? Ideal buffer: cin=0,Matched Air-Gap Reference Capacitors,Simp

4、le Capacitor Divider (con.),matched air-gap reference capacitor,offset,signal,Capacitor Divider With Differential Excitation,Why modulate v+ and v- ? Ideal buffer: cin=0,Impedance divider with superposition:,Improved Capacitive Divider (cont.),no offset!,distortion,The capacitive Half -Bridge,Impeda

5、nce divider with superposition:,The capacitive Half Bridge (cont.),Simplify expression:,No offset， 2x signal increase,Parasitic Capacitances,Surface micromachined z-axis parallel-plate capacitor,Equivalent circuit,Cpp (x): nominal | plate sense capacitor Cf1 (x): fringe capacitance (varies with plat

6、e displacement) Cf2 :fringe capacitance between upper plate (connected to anchor plane) and lower plate slight dependence on x Cpu :parasitic capacitance from upper plate to substrate Cpl : parasitic capacitance from lower plate to substrate,Velocity Sensing,Fundamental current-voltage relationship

7、for a time-varying capacitor: Consider special case: v=vp =constant used in high-quality capacitance microphones,Velocity Sensing (cont.),Sense capacitors time variation: Parallel-plate sense capacitor with gap go : Harmonic motion:,Some Numbers,Surface micromachined capacitor:,Is this real?,noise i

8、n buffer amp,World Record Capacitive Position-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, 1860-1866,Full scale Corillis-induced displacement=20 Sense capacitance 1000fF Minimum

9、detectable capacitance change 12 zF =0.012 aF Nominal sense gap = 1.6 m Minimum displacement: 16 fm ! * Surface micromachining class audio frequency band,EE C245-ME C218 Fall 2003 Lecture 12,Is ADL Splitting Electrons?,At V+ =5V, the charge on the sense capacitor is: qs =c+ v+ =(1000fF)(5V)=5000fC N

10、umber of electrons at Minimum detectable change in sense charge: Minimum detected change in number of electrons:,电容变化 静电力,变间隙电容驱动器的基本理论Basic physics of Electrostatic Actuation,Two ways to change the energy: 1.Change the charge q 2.change the separation x Note: we assume that the plates are supported

11、 elastically ,so they dont collapse .,Charge-Control Case (cont.),Stored energy:,Force (attractive ,internal):,Voltage:,Independent of the gap!,constant,Electrostatic Force (Voltage Control),Find co-energy in terms of voltage,Variation of co-energy with respect to gap yields v.s. force:,Variation of

12、 co-energy with respect to voltage yields charge,as expected,Linearizing the Voltage Square-Law,Polarize the capacitor by applying a DC offset voltage VP together with a (small) signal voltage Vsig (t) VP,DC offset,neglect (small),The Differential Electrostatic Actuator,Net force on suspended center

13、 electrode is the difference,Parallel Plate Capacitive Nonlinearity,Example: laterally driven spring suspended plate (eventually with balanced electrodes ) Nomenclature,Conductive structure,electrode,Value,AC or signal component (lower case variable subscript),DC Component (upper case variable: uppe

14、r case subscript),Parallel Plate Capacitive Nonlinearity,Example: clamped-clamped laterally driven beam with balanced electrodes Expression for,Expand the Taylor Series further,Conductive structure,electrode,Parallel plate Capacitive Nonlinearity,Parallel Plate Capacitive Nonlinearity,Retaining only

15、 terms at the drive frequency: These two together mean that this force acts against the spring restoring force! A negative spring constant since it derives from VP we call it the electrical stiffness, given by:,Drive force arising from the input excitation voltage at the frequency of this voltage,Pr

16、oportional to displacement,900 phase-shifted from drive, so in phase with displacement,Electrical stiffness ,Ke,The electrical stiffness ke behaves like any other stiffness It affects resonance frequency:,Frequency is now a function of dc-bias Vp1,Can One Cancel Ke with Two Electrodes?,What if we do

17、nt like the dependence of frequency on VP ? Can we cancel KC via a differential input electrode configuration ? If we do a similar analysis for Fd2 at Electrode 2:,Subtracts from the Fd1 term ,as expected,Add to the quadrature term Kcs add, no matter the electrode configuration!,The capacitive Half

18、-Bridge,Impedance divider with superposition:,The capacitive Half Bridge (cont.),Simplify expression:,Electrostatic force:,Electrostatic Force (Cont.),Output voltage is proportional to the displacement (for xgo),DC and 2w terms,Electrostatic Spring Constant ke,note direction: spring applies force op

19、posite to displacement,Both DC and 2w components : use square wave excitation to yield constant ke,Graphical Solution for Plate Stability,Plot normalized electrostatic and spring forces vs. normalized displacement 1-(g/go),So why are electrostatic actuators important in MEMS ,anyway?,Easy to make in

20、 micromachining processes ,since conductors and air gaps all thats needed Energy conserving only parasitic energy loss through i2R losses in conductors and interconnects Pull-in phenomenon can be exploited to make a hysteretic actuator simplifies control Multiple plate structures (combs ,3D) can be

21、used to tailor the force (displacement voltage) function Scaling of the electrostatic force is favorable due to Paschens curve Same structure can be used for position sensing,Paschens Curve,叉指驱动器的理论模型(Electrostatic comb drive ),Use of comb-capacitive tranducers brings many benefits Linearizes voltag

22、e-generated input forces (Ideally) eliminates dependence of frequency on dc-bias Allow a large range of motion,Electrostatic Force: a First Pass,Stator (fixed electrode) Rotor (notbut moving),Gap=g, thickness=t L=finger length X=overlap length,First-Pass Electrostatic Force (cont.),Neglect fringing

23、fields Parallel-plate capacitance between stator and rotor,Independent of x！,Relative Force for Surface Microstructures,Comb drive (x-direction) (V1 =V2 =VS =1V),Differential | plate (y-direction) (V1 =0V,V2 =1V),| plate wins bigfor surface MEMS,Gap=g=1 Thickness=t=2 Finger length=L=100 Overlap leng

24、th x=75,Comb Drive Force : a Second Pass,Energy must include capacitance between the stator and rotor and underlying ground plane, which is typically biased at the stator voltage Vs why？,Comb Drive Force with Ground plane Correction,Finger displacement changes capacitances from stator and rotor to t

25、he ground plane modifies the electrostatic energy,Capacitance Expressions,Consider case where Vr =VP =0V Csp =depends on whether or not fingers are engaged,Capacitance per length unit,Simulation (2D Finite Element),20-40% reduction of Fe,Vertical Force (Levitation),consider Vr =0V as shown:,Levitati

26、on Force,“electrical spring const.”,constant,Levitation force adds to the mechanical spring constant in the z direction increases the resonant frequency,Vertical Resonant Frequency,Must account for electrical springs in finding MEMS resonant frequencies Comb (x-axis) Ke =0 Comb (z-axis) Ke 0 Paralle

27、l plate Ke 0,第4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,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

28、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

29、的横向应力相对于晶体主轴系中的方向余弦 。,Relative resistance change can be expressed by the longitudinal and transverse piezoresistive coefficients Piezoresistors are often aligned to the wafer flat of (100) wafers, which is in the 110 direction. Senturia,p.473 provides the result of coordinate transformations:,Silico

30、n piezoresistive coefficients,Function of type ,doping, and temperature Longitudinal and transverse coefficients in110 direction,n-type 11.7 -102.2 53.4 -13.6 P-type 7.8 6.6 -1.1 138.1 Units -cm, 10-1Pa-1 values are at T=25 0C,n-type P-type,一般地，当晶面为(100)时，有,表7-9 P型压电阻在各方向的压阻系数,Piezoresistor Placemen

31、t,Bulk micromachined diaphragm pressure sensor,电阻变化的read-out,公式?,举例计算电阻的变化导致电压的变化,第4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 Effect,Several views of an -quartz crystal,Origin of Piezoelectric Effect,For ra, the electric field at the point P is:

32、 The potential and electric field appear as if the charges are coincident at their center of gravity (point O),Origin of Piezoelectric Effect,Assume the applied force F causes the line OD to rotate counter clockwise by a small angle This strain shifts the center of gravity of the three positive and

33、negative charges to the left and right, respectively A dipole moment, p=qr, is created which has an arm (r) of: p=qr qa33/2 Assuming the crystal contains N such molecules per unit volume, each subject to the same strain , the polarization (or dipole moment per unit volume) is:,polarization,strain,Or

34、igin of Piezoelectric Effect,For sufficiently small deformations , polarization (p) is linearly related to the strain (s) by: p=gs where g is the piezoelectric voltage coefficient. Converse Piezoelectric Effect When a piezoelectric crystal is placed in an electric field, positive and negative ions a

35、re pushed in opposite directions and a dipole tends to rotate to align itself with the electric field. The resulting motion gives rise to strain s that is proportional to electric field E S=dE where d is the piezoelectric charge coefficient.,Anisotropic Crystal Properties: Generalized Stress-Strain,

36、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, longitudinal axis. Since we have 6 components of stress (T) and 6 components of strain (S), 36constants mus

37、t be used to describe behavior in the general case. Crystal symmetry (e.g. trigonal, hexagonal) greatly reduces the number of independent constants.,Anisotropic Crystal Properties: Generalized Stress-Strain,For small deformations, stress (T) and strain (S) are related though the compliance matrix (s

38、) Conservation of energy requires sij=sji. Performing rotations based upon trigonal symmetry considerations, the compliance matrix reduces to 6 independent coefficients:,Quartz has threefold symmetry, physical properties repeat every 1200. Quartz is also symmetric about the x-axis,Anisotropic Crysta

39、l Properties: Generalized Stress-Strain,Recall that the strain (S) is related to the electric (E) by the piezoelectric charge coefficient matrix (d),Applying the symmetry conditions for quartz, the piezoelectric strain matrix (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 V