Abstract: In this paper, a novel method has been developed to eliminate the tensile or compressive residual stresses effects that have been created due to the fabrication sequence. In the developed model, the tensile or compressive residual stresses in the fixed-fixed end type MEM switches have been eliminated by using of two piezoelectric layers that have been located on the upper and lower surfaces of switch. The nonlinear governing differential equation of the fixed-fixed end type MEM switch has been derived by considering the piezoelectric layers and residual stresses. The pull-in voltage and the switching time have been calculated for different applied voltage to the piezoelectric layers. The results show that the both tensile and residual stresses can be eliminated by means of a small applied voltage to the piezoelectric layers

Rezazadeh Ghader

Tahmasebi Ahmadali

In this paper, a novel method has been developed to eliminate the tensile or compressive residual stresses effects that have been created due to the fabrication sequence. In the developed model, the tensile or compressive residual stresses in the fixed-fixed end type MEMS switches have been eliminated by using of two piezoelectric layers that have been located on the upper and lower surfaces of switch. The nonlinear governing differential equation of the fixed-fixed end type MEMS switch has been derived by considering the piezoelectric layers and residual stresses. The pull-in voltage and the switching time have been calculated for different applied voltage to the piezoelectric layers. The results show that the both tensile and residual stresses can be eliminated by means of a small applied voltage to the piezoelectric layers

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Sensors & Transducers Magazine (S&T e-Digest), Vol.66, Issue 4, April 2006, pp.534-542     Sensors & Transducers ISSN 1726-5479© 2006 by IFSAhttp://www.sensorsportal.com        Eliminating of the Residual Stresses Effect in the Fixed-Fixed End Type MEMS Switches by Piezoelectric Layers  Rezazadeh Ghader1, Tahmasebi Ahmadali21, 2Mech. Eng. Dept. Urmia University, Urmia, Iran E-mail: 1 g.rezazadeh@mail.urmia.ac.ir   Received:  21 April 2006   Accepted: 24 April 2006   Published: 25 April 2006   Abstract: In this paper, a novel method has been developed to eliminate the tensile or compressive residual stresses effects that have been created due to the fabrication sequence. In the developed model, the tensile or compressive residual stresses in the fixed-fixed end type MEM switches have been eliminated by using of two piezoelectric layers that have been located on the upper and lower surfaces of switch. The nonlinear governing differential equation of the fixed-fixed end type MEM switch has been derived by considering the piezoelectric layers and residual stresses. The pull-in voltage and the switching time have been calculated for different applied voltage to the piezoelectric layers. The results show that the both tensile and residual stresses can be eliminated by means of a small applied voltage to the piezoelectric layers.    Keywords: MEM switches; piezoelectric; residual stresses; pull-in voltage; switching time  __________________________________________________________________________________  1. Introduction  Recently, there have been many investigations on micro-electromechanical (MEM) switches to develop low power consumption, low cost and ultra-low loss switches with a goal of replacing traditional electronic switching components (such as pin diodes). The development of capacitive MEM switch was phenomenal. The advent of monolithic integration of the micromechanical structure with electronics requires the development of simulation and analysis tools that can allow coupled circuit and micromechanical simulation [1-3]. Pull-in phenomenon is a discontinuity related to the interplay of the elastic and electrostatic pressures. The determination of the pull-in voltage and position requires the solution of a coupled electrostatic-elastic system. Because of miniaturization and electromechanical coupling, some new phenomena and factors, such as residual stress in thin films and so on, become comparatively more important. An understanding of the thin film material properties in  534Sensors & Transducers Magazine (S&T e-Digest), Vol.66, Issue 4, April 2006, pp.534-542 a MEMS process is essential for detailed design and analysis. It is also important for process control as these properties can vary from lot to lot and across a single wafer. Important properties include stress gradient and residual stress. Many deposition parameters including doping levels, thermal conditions, impurities, grain growth and orientation influence these properties [4-6].  As the fixed-fixed beams deflect, the axial force through the beam causes stiffening to occur, changing the deflection of the beam.  Residual stress is an important part of this axial force, the deflected shape of the beam depends strongly on the magnitude of the residual stress and cause the parameters of the switch vary lot to lot. Depending on the fabrication sequences of the MEM switches, the residual stresses can be tensile or compression. Tensile residual stress stiffens micro beams but conversely compressive residual stress softens them. The residual stresses can make forces in the section of the beam as residual forces. These residual forces can influence on the characteristics of the switches such as pull-in voltage and switching time where they are can be desired and undesired, either. In this paper, is tried to eliminate the residual forces by means of the piezoelectric layers. It is assumed that there is no stress gradient along the height of the beam meanwhile the stresses are uniform in that direction. And because of the much smaller the height of piezoelectric layers than the height of the beam, it is assumed there is no residual forces in the piezoelectric layers.   2. Mathematical Model for the Fixed-Fixed Electrostatic MEM Switches with Piezoelectric Layers  Now, as it is shown in the Fig. 1, assume a fixed-fixed end type beam with thickness , width b , lengthhL , density ρ and isotropic with Young’s modulus E  that has a pair of piezoelectric layers bonded on its surfaces. Each piezoelectric layer has thickness , density1h 0ρ , Young’s modulus and equivalent piezoelectric coefficient for one-dimensional problem pE31e . Suppose that x is the coordinate along the length of the beam with its origin at the left end, and  is the deflection of the beam, defined to be positive downward. The piezoelectric layers are located throughout the beam length.  )(xw   Fig. 1. Schematic of fixed-fixed MEM switches with piezoelectric layers.   For beams,  is usually large enough to neglect the shear deformation and the rotary inertia. So, the longitudinal displacement of the beam may be written as: hL / xtxwztxu∂∂−=),(),(   (1) where and are the displacements in the transverse z-direction and longitudinal x-direction, respectively. The amplitude of the strain),( txw ),( txuxε , stress xσ , and the moment  in the beam due to  are given yMw 535Sensors & Transducers Magazine (S&T e-Digest), Vol.66, Issue 4, April 2006, pp.534-542  22xwzxux ∂∂−=∂∂=ε      ,    22xwEzx ∂∂−=σ         ,  ∫− ∂∂−==2/2/ 22hh xy xwEIdzzM σ   (2) where . Residual stress, due to the mismatch of both thermal expansion coefficient and crystal lattice period between substrate and thin film, is unavoidable in surface micromachining techniques, so that accurate and reliable data of residual stress is crucial to the proper design of the MEMS devices concerned with the techniques [1, 2]. Therefore, the residual stress is an attractive research topic in the development of the Microsystems Technology (MST). Considering the fabrication sequence of MEM switches, the residual stress is very important and inevitable to the device. Where 3)12/1( bhI =)1(ˆ 0 νσσ −=  for fixed-fixed beam, where 0σ is the biaxial residual stress [3].  Therefore, the total amplitude of the stresses in the beam section can be written as:  σσ ˆ22+∂∂−=xwEzx   (3) The stress in the piezoelectric layer accounting for electro-mechanical effects can be expressed as [7]:   zpx EexwzE 3122−∂∂−=σ ,  (4)  where is the electric field in the piezoelectric layer. If the piezoelectric layers are placed symmetrically and poled in the transverse direction of the beam, then applying equal voltages  to the upper and lower piezoelectric layers will induce tensile or compressive mechanical stresses on the beam if complete bonding is assumed, as pointed out by [7]. Meanwhile, we have the following equations as: zEpV  PhbdzbxwEzhh==+∂∂−∫+−σσ ˆ)ˆ(2/2/ 22  (5)   FVebeEbhdzbEexwzEdzbEexwzE pzhhh zphhh zp=−=−=−∂∂−+−∂∂− ∫∫+++−−− 313112/2/ 31222/2/ 312222)()( 11 (6)  where . Such tensile or compressive force induced by the piezoelectric layer, due to the electric field, can be used to eliminate the residual stresses in a MEM switches. 1/ hVE pz = The governing equations for the beam subjected to a residual compressive and tensile piezoelectric force under the dynamic conditions can be expressed as:   )()2()())2/(( 221/224411 xqtwbhbhxwFPxwhhhbhEEI p =∂∂++∂∂−+∂∂++ ρρ   (7)  whereas 202)(2)(wgVbxq a−=ε is the distributed non-uniform electrostatic pressure that applied to the lower surface of the beam, is the initial gap, 0g ε  is Permittivity of air or the dielectric coefficient and  is the actuation voltage that impose to the electrostatic area. aV 536Sensors & Transducers Magazine (S&T e-Digest), Vol.66, Issue 4, April 2006, pp.534-542  3. Solution of the Governing Equations  By using the method of separation, the general solution for the homogenous part of the above equations can be noted as:   tiexWtxw ω)(),( =  (8)  By substituting Eq.(8) into Eq.(7), the governing equation can be written as:   0)2()())2/(( 21/224411 =+−−+++ WbhbhdxWdFPdxWdhhhbhEEI p ωρρ   (9)  Then we have:   )cos()sin( 242321 11 xAxAeAeAWxx λλλλ +++= −            FP ≥  )cos()sin( 141321 22 xAxAeAeAWxx λλλλ +++= −            PF >  (10)  (11) where   ⎪⎪⎩⎪⎪⎨⎧++−−∆=++−+∆=))2/((2)())2/((2)(112111hhhbhEEIPFhhhbhEEIPFppλλ          21/112 )2))(2/((4)( ωρρ bhbhhhhbhEEIFP p ++++−=∆    For the beam of Fig.1, there are four boundary conditions. The fixed-fixed conditions can be written as:  At        0=x⎪⎩⎪⎨⎧=∂∂=00xww   and at ⎪⎩⎪⎨⎧=∂∂==00xwwLx   (12) By imposing the boundary conditions into the Eqs. (10), (11), leads to the below equations, which can be expressed in the following form:   ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−⇒= −0000)sin()cos()cos()sin(010110~~~4321222211222111111AAAALLeeLLeeARLLLLλλλλλλλλλλλλλλλ             FP ≥    (13) and   537Sensors & Transducers Magazine (S&T e-Digest), Vol.66, Issue 4, April 2006, pp.534-542  ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−⇒=−−0000)sin()cos()cos()sin(010110~~~4321111122111222222AAAALLeeLLeeARLLLLλλλλλλλλλλλλλλλ              PF >  (14) For non-trivial solution of the vector [ , the following conditions are imposed: ]Τ4321 AAAA  0)~det( =R              PF ≤  0)~det( =R               PF >  Theoretically, this will give the eigen frequencies of the system. By supposing , we can gain the other constant coefficients. Finally, the solution of the fixed-fixed piezoelectric MEM switch  can be obtained. Hence, by applying the virtual work principle we can model the switch into a one degree of freedom model. 11 =A),( txw  2/2021)2/1()2)(2/1(LxLeqwdxwbhbhm=∫ ′+=&&ρρ  (15) Therefore:   2ωeqeq mk =  (16) where is the equivalent mass, is the equivalent stiffness and is the natural frequency of the beam. We can estimate the pull-in voltage and switching time in a fixed-fixed MEM switch by using the following equations that was govern by Rebeiz and Barker [], where the damping coefficient is assumed to be zero: eqm eqk ω  AgkV eqPI ε278 30=   ωaPIVVt 67.3= ,  (17)   (18) where  is the pull-in voltage in fixed-fixed MEM switch and t  is the switching time in a fixed-fixed MEM switch. PIV  4. Numerical Simulation and Discussions  As an illustration, a polysilicon beam with PZT-4 piezoelectric layers attached is considered with the geometric and material constants listed in Table 1. The non-dimensional frequency of the beam is defined as follows: ))2/(/()2( 11421 hhhbhEEILbhbh p ++′+= ωρρω .   538Sensors & Transducers Magazine (S&T e-Digest), Vol.66, Issue 4, April 2006, pp.534-542 First, consider the case of a beam with 0=F  and 0=P , implying that no piezoelectric and no residual force. Therefore, the obtained non-dimensional natural frequencies are 4.7300 and 7.8532 for the first and second natural modes, respectively. These values compare well with the values of 4.7300 and 7.8532 provided by D. Young and R. P. Felgar [10] for the first and second natural frequencies of a fixed-fixed beam, respectively. In this case, by using of the Eqs.(17,18), we calculate the pull-in voltage and switching time that are equal to 25.01 (V) and 14.39(µs) when the actuation voltage that is the same as the pull-in voltage, respectively. We declare them the basic values and would obtain the ratio of pull-in voltage and switching time respect to them.  Polysilicon is typically deposited by LPCVD using thermal decomposition of silane (SiH4). As with all deposition processes, the microstructure of LPCVD polysilicon films is dependent on the deposition conditions. In fact, the deposition temperature for the individual layers alternates, which produces alternating tensile or compressive residual stresses [9]. Tensile stresses may be produced by using of PECVD method, too [10]. Therefore, the elimination of the two kinds of stresses is considered. First, we try to eliminate the compressive residual stress that is about )(2.50 MPa−=σ  by means of inducing the tensile force in the piezoelectric layers.   Table 1. Geometrical and material properties of the MEM switch and piezoelectric layer.   MEM Switch piezoelectric layer Length 600 µm 600 µm Width 50 µm 50 µm Height 3 µm 0.01 µm Young’s modulus 165 (GPa) 78.6 (GPa) Poisson’s ratio 0.29 0.3 Mass density 2331 (Kg/m3) 7500 (Kg/m3) 31e  - -9.29 0g (initial gap) 1 µm  ε (Permittivity of air) 8.8541878×10-12 (F/m)    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.24.34.44.54.64.74.84.95Applied Voltage to The Piezoelectric Layers (V)None-Dimensional FrequencyFig. 3. Applied voltage to the piezoelectric layers versus first none-dimensional natural frequency. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.850.90.9511.051.1Applied Voltage to The Piezoelectric Layers (V)The Ratio of The Pull-in Voltage Fig. 4. Applied voltage to the piezoelectric layers versus the ratio of pull-in voltage.   539Sensors & Transducers Magazine (S&T e-Digest), Vol.66, Issue 4, April 2006, pp.534-542 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9920.9940.9960.99811.0021.0041.006Applied Voltage to The Piezoelectric Layers (V)The Ratio of The Switching Time Fig. 5. Applied voltage to the piezoelectric layers versus the ratio of switching time  0 100 200 300 400 500 600-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10Discretization in x direction (µm)First Mode Shape (µm)Vp=0Vp=0.2Vp=0.5Vp=0.8Vp=1 Fig. 6. First mode shape of the beam due to different applied voltages in piezoelectric layers.   As it is shown in the Figs. (3-6), the compressive residual stresses soften the fixed-fixed MEM switches. So the pull-in voltage and switching time are low than to the ideal cases and by imposing the tensile piezoelectric force, the pull-in voltage and switching time are increased. Then is tried to eliminate the tensile residual stresses that is about )(2.50 MPa=σ , by means of inducing the compressive force in the piezoelectric layers that means by changing the direction of the electric field. In this case, if assume that  is represented as the residual force and F P  is represented as the piezoelectric force, the governing equations are the same as before. As are observed in the Figs. (7-10), the tensile residual stresses stiffen the fixed-fixed MEM switches. So the pull-in voltage and switching time are more than ideal cases and by imposing the compressive piezoelectric force, the pull-in voltage and switching time are decreased. Finally, as it is shown in the Figs. (3-10), because of the tensile and the compressive residual stresses have the same value in our analysis, the natural frequency, the pull-in voltage and the switching time are become equal to their values of a simple Fixed-Fixed MEM switch when the residual force is equal to the piezoelectric force that is happening at . That means the residual stresses by using of piezoelectric layers successfully are eliminated. )(597.0 VVp = 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14.54.64.74.84.955.1Induced Voltage to The Piezoelectric Layers (V)None-Dimensional FrequencyFig. 7. Applied voltage to the piezoelectric layers versus none-dimensional frequency.  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.90.9511.051.11.15Applied Voltage to The Piezoelectric Layers (V)The Ratio of The Pull-in Voltage Fig. 8. Applied voltage to the piezoelectric layers versus the ratio of pull-in voltage.   540Sensors & Transducers Magazine (S&T e-Digest), Vol.66, Issue 4, April 2006, pp.534-542 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9940.9960.99811.0021.0041.0061.008Applied Voltage to The Piezoelectric Layers (V)The Ratio of The Switching Time Fig. 9. Applied voltage to the piezoelectric layers versus the ratio of switching time  0 100 200 300 400 500 600-0.7-0.6-0.5-0.4-0.3-0.2-0.10Discretization in x direction (µm)First Mode Shape (µm)Vp=0Vp=0.2Vp=0.5Vp=0.8Vp=1Fig. 10. First mode shape of the beam due to different applied voltages in piezoelectric layers.    5. Conclusion  In this paper, as it has observed, the elimination of the both tensile and compressive residual stresses in fixed-fixed MEM switches can be done successfully, by using of piezoelectric layers with small applied voltage. However, the eliminating of such residual stresses may be not seems good. In fact, the tensile residual stress stiffens the fixed-fixed MEM switches and causes the high pull-in voltage and switching time that seem undesirable. Meanwhile, the compressive residual stress softens the fixed-fixed MEM switches and decrease the pull-in voltage and switching time that seem desirable. Therefore, by means of the piezoelectric layers we can induce compressive force to decrease the pull-in voltage and switching time or can induce the tensile force to increase the pull-in voltage small applied voltage to the piezoelectric layers.   References  [1]. T. Mukherjee, G. K. Fedder, and J. White, Emerging simulation approaches for micromachined devices, IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., Vol.19, pp. 1572–1589, Dec. 2000. [2]. S. D. Senturia, N. Aluru, and J. White, Simulating the behavior of MEMS devices, IEEE Computat. Sci. Eng., Vol. 4, no. 1, pp. 30–43, 1997. [3]. S.D.Senturia, CAD challenges for microsensors, microactuators, and microsystems, Proc. IEEE, Vol. 86, pp. 1611–1626, 1998. [4]. Kamins, T. I., Design properties of polycrystalline silicon, Sensor and Actuators A, Vol. A23,  pp. 817-824, 1990. [5]. Adamaczewska, J., and Budzynsky, T., Stress in chemically vapour-deposited silicon films, Thin Solid Films, 113 (4), 271, 1984. [6]. Maier-Schneider, D., Koepurueluelue, A., Ballhausen Holm, B., and Obermeier, E., Elastic properties and microstructure of LPCVD polysilicon films, Journal of Micromechanics and Microengineering, Vol. 6, Issue 4, pp. 436-446, December 1, 1996. [7]. E.F. Crawley and J. de Luis, 1987, Use of Piezoelectric Actuators as Elements of Intelligent Structures, AIAA Journal, 25(10): 1373-1385. [8]. D.Young, and R. P. Felgar, Jr., Tables of Characteristic Functions Representing Normal Modes of Vibration of a Uniform Beam. The University of Texas Publication No. 4913, July 1. 1949. [9]. H. Kahn, A. H. Heuer, Thermal stability of residual stresses and residual stress gradients in multilayer LPCVD polysilicon films, Journal of Ceramic Processing Research, Vol. 3, No. 1, pp. 22-24, 2001.  541Sensors & Transducers Magazine (S&T e-Digest), Vol.66, Issue 4, April 2006, pp.534-542 [10]. R. Ghodssi, X. Zhang, K-S Chen, S. M. Spearing, and M. A. Schmidt, Residual Stress Characterization of Thick PECVD Oxide Film for MEMS Application, The 46th International Symposium of the American Vacuum Society, Seattle, WA, 10/25-29, 1999.  ___________________                                                 2006 Copyright ©, International Frequency Sensor Association (IFSA). All rights reserved. (http://www.sensorsportal.com)   542

Eliminating of the Residual Stresses Effect in the Fixed-Fixed End Type MEMS Switches by Piezoelectric Layers

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Eliminating of the Residual Stresses Effect in the Fixed-Fixed End Type MEMS Switches by Piezoelectric Layers

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