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Engineering, 2010, 2, 55-59 doi:10.4236/eng.2010.21007 lished Online January 2010 (http://www.SciRP.org/journal/eng/). Copyright © 2010 SciRes. ENGINEERING 55 Pub Analysis for Pull-In Voltage of a Multilayered Micro-Bridge Driven by Electrostatic Force Yu LIU1,2,3, Guochao WANG1,2,3, Hongyun YANG1,2,3 1Automobile College, Chongqing University of Technology, Chongqing, China 2Key Laboratory of Automobile Parts & Test Technique in Chongqing, Chongqing, China 3Chongqing Engineering Research Center for Automobile Power System and Control, Chongqing, China Email: liuyu_cq@126.com Received August 8, 2009; revised September 4, 2009; accepted September 12, 2009 Abstract A trial solution for bending deflection of a multilayered micro-bridge subject to a voltage induced load is presented. The relation between the applied voltage and the displacements of the micro-bridge in the pull-in state is analyzed by energy method. Furthermore, two analytical expressions about normalized displacement and pull-in voltage are carried out. It’s proved that the value of normalized displacement is not influenced by residual stress if axial and shear deformation is ignored. Finally, the theoretical results are compared with that of FEM, and they show good agreement. Keywords: MEMS, Electrostatic Actuation, Multilayered Micro-Bridge, Trial Solution, Energy Method, Pull-In Voltage 1. Introduction Moving part is the most frequently used one in MEMS structures. There are various principles that can be used to drive the moving part, including electrostatic, piezoelectric, thermal, magnetism, etc. Compared with the others, the method of electrostatic force drive is more attractive [1,2] because of its larger force caused by micro-effect and non-contact, which is beneficial to high precision. Further- more, the process of MEMS devices driven by electro- static force is compatible with IC process. Micro-bridge structure driven by electrostatic force is familiar in MEMS devices, such as changeable capaci- tors, RF switches, micro-resonators, pressure sensors, and so on. On one hand, it is a common structure being used to obtain the mechanical parameters of film, just like Young’s Modulus, residual stress, yield strength and bending strength [3–5]. But on the other hand, MEMS devices with micro-bridge structure are often with low reliability and worse quality, which is induced by pull-in phenomena. So it’s worthy to research the pull-in phe- nomena of micro-bridge structure subject to a voltage induced load, which is important to MEMS design and optimization. For example, drive voltage must be less than pull-in voltage for most structures driven by elec- trostatic force, while it’s opposite for DMD and self- measure unit of micro-accelerometer [6]. Although many people study the pull-in phenomena of the monolayered micro-bridge structure driven by elec- trostatic force, and get some valuable conclusions, they, however, cannot be applied to the multilayered micro- bridge structure. So some people [7] devote themselves to research on the pull-in phenomena of the multilayered micro-bridge structure. Whereas the analytical expres- sion is complicated, this depends on the introduction of the assumptions and the form of the trial solution. This paper presents a trial solution for bending deflec- tion of a multilayered micro-bridge subject to a voltage induced load. The relation between the applied voltage and the displacements of the micro-bridge in the pull-in state is analyzed by energy method. Furthermore, two analytical expressions about normalized displacement and pull-in voltage are carried out. It’s proved that the value of normalized displacement is not influenced by residual stress if axial and shear deformation is ignored. Finally, the theoretical results are compared with that of FEM, and they show good agreement. 2. Model The micro-bridge with a rectangular cross-section is made of n layer materials, illustrated in Figure 1. Here we assume the top of the beam is a conductor, while the others are dielectric. The electrode under the beam is Y. LIU ET AL. 56 Figure 1. Schematic picture of a multilayered micro-bridge subject to a voltage induced load. Figure 2. Deformation of an infinitesimal line element dx. covered with the insulated material, which is fixed on the substrate. It’s not difficult to imagine that micro-bridge will be bended when the voltage is added to the beam and the electrode. Let’s explain the meanings of some symbols firstly, where l is the length of the beam. , , i hi Eri are the thickness, Young’s Modulus and relative permittivity of the i-layer material respectively. , 0 h0r are the thick- ness and relative permittivity of the insulated layer above the electrode respectively. V is the voltage added to the beam and the electrode. h is the gap distance between the bottom of the beam and the top of the electrode when . 0V Set symmetric axis of beam’s cross section as z axis, and positive direction is upwards. Set the equivalent neutral axis as y axis, and its position is needed to be confirmed. Then x axis is the perpendicular axis of yz plane and through the origin. The distance from the equivalent neutral axis to the bottom of the beam is. It’s clear that eff z 1 1 2 n eff i i z As shown in Figure 2, according to the plane cross- section assumption on pure bending, we calculate the strain i of the fiber bb while, which is in the -layer and the primary length as well as the gap distance to neutral surface are dx, respectively. 0V i i z i i z where is curvature radius of the neutral surface. Based on the assumption on pure bending, that is, there is no normal stress between the longitudinal fibers; we calculate the stress i of the fiber bb according to Hooke’s law. Of course, the deformation is at the elastic deformation period. i iii i z EE Uncountable micro internal force idA composes a system of parallel force in space, which can be simplified as three internal force components. What we’re con- cerned about is the component parallel to x -axis, namely axial force. Based on the equilib- rium of static equilibrium i A N dA 0XN and the follow- ing formula, 12 12 112 12 1121 112 2 1112 22 1112 2 2 1 1 n n effeffn eff effeffn eff i A nn AA A nn n AA A hzhh zhhh z nnn zhzhhhz NdA dA dAdA E z dAEzdAEzdA E bzdzEbzdzEbzdz We get 1 11 1 () 2 j nj j ji ji eff n ii i h Eh h z Eh Therefore, 0eff zz 1 () 1,2,, i ijeff j zhzi n where are the coordinates of the micro-bridge’s top and bottom surface along -axis, while 0n zz, z11n zz ,, are that of layer-layer interface respectively, as shown in Figure 1. 3. Solution h due to the different Young’s Modulus of n layer materials. Here we choose a trial solution for bending deflection of Copyright © 2010 SciRes. ENGINEERING Y. LIU ET AL. 57 a multilayered micro-bridge subject to a voltage induced load as following [8]: 2 2 ()(1 cos)2 sin x x wx AA ll where A is the undetermined coefficient and 2A is the maximum deflection of the micro-bridge. It’s acceptable to neglect the influence of axial and shear deformation on bending due to the small displace- ment assumption of the micro-bridge. 3.1. No Consideration of Residual Stress For this situation, the strain i of the i-layer material only includes flexural strainbi . That is, 22 22 42 cos ibi dwA x zz l dx l So the unit volume deformation energy and the de- formation energy of the i-layer material are described by the following formula: 2 1 2 ii uE i 1 2 0 2 i ii lz i siiii vz Eb Uudvdzdx Therefore, the total deformation energy of the mi- cro-bridge Us is equal to the summation of the deforma- tion energy of n layer materials. That is, 1 2 0 11 2 i i nn lz i ssi i z ii Eb U Udzdx 2 43 1 1 2 () ( 12 n ii i i blA Ezz l 3 ) Considering the influence of the insulated layers on the gap distance, we introduce the equivalent gap distance [9] h 01 1 01 (1 n rr rn hh h hh ) where iri h is the equivalent air gap distance of the i-layer insulated material. Then the model in Figure 1 is simplified as Figure 3, where is the equivalent thick- ness, which will be discussed in detail later. So the electrostatic energy can be described as [8] 2 0 2( 2) e Vb l U hh A where 0 is the permittivity of vacuum. Therefore, the energy function is described as [10] s e UU Because the value of the electrostatic force is in in- verse proportion to the square of distance, and the distri- bution of electric field is influenced by structural dimen- sion as well as position, so the issue of bending equilib- rium stability should be considered. It exists when the electrostatic force is equal to the elastic force. If the voltage V exceeds the critical value, the equilibrium state is destroyed and the micro-bridge will contact with the fixed plane, which is called pull-in phenomena. The minimum critical voltage is called pull-in voltage, marked as VPI. Here, 0 A and 2 20 A (1) Mark the maximum deflection of the micro-bridge under VPI as 2Acr, then 2 3 ) ~ 2 ~ ( ~ 2 )() 2 ( 6 2 2 3 1 3 1 4 crii n i i cr Ahhh V zzE l A (2) 2 5 ) ~ 2 ~ ( ~ 2 3 )() 2 ( 6 122 2 3 1 3 1 4 crii n i iAhhh V zzE l (3) So 0.2 cr A h (4) And the normalized displacement 20.4 cr A h 3.2. Consideration of Residual Stress For this situation, the strain i of the i-layer material includes not only flexural strain bi but also the strain Figure 3. The equivalent model of a multilayered micro- bridge. Copyright © 2010 SciRes. ENGINEERING Y. LIU ET AL. 58 induced by the residual stress i of the i-layer material. That is, 22 22 42 cos ii ibi ii dwA x zz EE ldx l i i E Similarly, the deformation energy of the micro-bridge 1 2 0 1 2 2 433 1 11 2 2 () () 12 2 i i nlz i si z i nn ii ii i ii i Eb Udzdx blh blA Ez z l E (5) Because the second part of Formula (5) is not the function of the undetermined coefficient A, so the results from Formula (1) are same to that of Subsection 3.1. It’s clear that the value of normalized displacement is not influenced by residual stress if axial and shear de- formation is ignored. Based on Equations (2) and (4), we get 333 1 1 4 0 () 48.290 n ii i i PI hEzz Vl (6) For the multilayered micro-bridge with a rectangular cross-section, the area moment of inertia )( 3 ~3 0 32 0 zz b dzbzIn z z n and the flexural stiffness n i iii n i z zizzE b dzbzEIE i i1 3 1 3 1 2)( 3 ~~ 1 then the equivalent Young’s Modulus 3 0 3 1 3 1 3)( ~ ~~ ~ zz zzE I IE E n n i iii and the equivalent thickness 1 13 333 0 12 () 4() n Izz b Therefore, the formula (6) is simplified as 33 4 0 12.0725 PI E h Vl (7) It’s clear that the multilayered micro-bridge can be considered as a monolayered micro-bridge if only the equivalent Young’s Modulus and the equivalent thickness E are carried out. In the above discussion, we assume the top of the beam is a conductor while the others are dielectric. If the conductor is not on the top of the micro-bridge, or more than one layer is conductor, the equivalent gap distance will differ. h 4. Validation All the above deduction is based on a given trial solution for bending deflection. Are they correct? How about the accuracy? Now we try to prove it by simulation. Firstly, some examples relating to the monolayered micro-bridges are considered. For the monolayered mi- cro-bridge given in paper [11], the value of VPI is 32.5V, and the simulated values by ANSYS as well as Intellisuite are 37.6V, 39.5V respectively, which are presented in paper [11]. Whereas the analytical value based on the Formula (7) is 34.2V, which is more close to the simu- lated values. And for that given in paper [12], the simu- lated value of VPI is 40V, while the analytical value based on the Formula (7) is 40.99V. The error is only 2.48%. Secondly, a multilayered micro-bridge with a rectan- gular cross-section is considered. It’s made of two layer materials. And the electrode under it is covered with 0.5 m Si3N4. Model 1: the top material is gold. And the other is Si3N4. Model 2: the top material is Si3N4. And the other is gold. Table 1. Geometry and material parameters of the bi-lay- ered micro-bridge. Item Symbol (unit) Model 1 Model 2 length l(m ) 400 width b(m ) 50 gap distance (V=0) h(m ) 2.0 thickness of the insulated layer 0 h(m ) 0.5 1 h(m ) 2.0 0.5 thickness 2 h(m ) 0.5 2.0 relative permittiv- ity 0r =1r 8.0 1 E(GPa ) 210 57 Young’s Modulus 2 E () GPa 57 210 Table 2. PI V, β of the bi-layered micro-bridge. ItemModel Analytical values Simulated valuesError model 137.9671V 40.2187V -5.60% VPI model 231.9798V 33.3438V -4.09% model 10.4069 -1.70% model 1#0.3933 1.70% model 20.3972 0.70% model 2# 0.4 0.4132 -3.19% Note: # Residual stress of the gold layer is 100 MPa. Copyright © 2010 SciRes. ENGINEERING Y. LIU ET AL. Copyright © 2010 SciRes. ENGINEERING 59 The geometry and material parameters of the bi-layered micro-bridge are listed in Table 1. The micro-bridge will be bended when subject to a voltage induced load. [2] G. Li and N. R. Aluru, “Efficient mixed-domain analysis of electrostatic MEMS,” ICCAD, pp. 474, 2002. [3] T. Y. Zhang, Y. J. Su, C. F. Qian, et al. “Micro-bridge testing of silicon nitride thin films deposited on silicon wafers,” Acta Mater, Vol. 48, pp. 2843, 2000. For Models 1 and 2, we get the simulated values of the pull-in voltage VPI and the normalized displacement by FEM. Table 2 lists the simulated performances and the analytical values of VPI and , as well as the error of them. It shows good correlation. So the trial solution of deflection is acceptable. [4] Y. Zhou, C. S. Yang, J. A. Chen, et al. “Investigation of Young’s modulus and residual stress of copper film micro- bridges by MEMS technology,” Modern Scientific In- struments, Vol. 4, pp. 45, 2003. [5] M. J. Wang, Y. Zhou, J. A. Chen, et al. “Measurements of elastic modulus and residual stress of nickel film by micro- bridge testing methods,” Electronic Components & Mate- rials, Vol. 23, No. 12, pp. 13, 2004. 5. Conclusions This paper presents a trial solution for bending deflection of a multilayered micro-bridge subject to a voltage induced load. The relation between the applied voltage and the displacements of the micro-bridge in the pull-in state is analyzed by energy method. Furthermore, two analytical expressions about normalized displacement and pull-in voltage are carried out. The theoretical results are com- pared with that of FEM, and they show good agreement. [6] H. Yang, M. H. Bao, S. Q. Shen, et al. “The displacement characteristics of the micromechanical structures driven by electrostatic force,” Journal of Fudan University (Natural Science), Vol. 38, No. 3, pp. 282, 1999. [7] H. Rong, Q. A. Huang, M. Nie, et al. “An analytical model for pull-in voltage of doubly-clamped multi-layer beams,” Chinese Journal of Semiconductors, Vol. 24, No. 11, pp. 1185, 2003. [8] D. H. Sun, Y. Q. Huang, W. Zheng, et al. “On the mod- eling methodology of MEMS system-level simulation,” Journal of Xiamen University (Natural Science), Vol. 40, No. 2, pp. 297, 2001. All results in this paper are based on the form of the selected trial solution. The better the selected trial solu- tion is, the higher the accuracy of solution is. Moreover, the axial and shear deformation is ignored, which maybe has influence on the stress and strain. Besides, in order to get an analytical expression of VPI, it’s difficult to con- sider the influence of some secondary effects, such as the marginal effect of electric field. However, this paper pro- vides an analytical model with high accuracy for a mul- tilayered micro-bridge driven by electrostatic force, which is beneficial to design, optimization and application of MEMS devices with the micro-bridge structure. [9] M. Nie, Q. A. Huang, J. H. Wang, et al. “Analysis of deflection and pull-in voltage of a multi-layer cantilever under an electrostatic load,” Chinese Journal of me- chanical engineering, Vol. 40, No. 8, pp. 72, 2004. [10] Y. Y. Wang, T. I. Kamins, B. Y. Zhao, et al. Polysilicon Film and its Application in IC (2nd Version), Science Press, Beijing, 2001. [11] C. S. Wang, W. B. Zhang, J. Fang, et al. “Research on coupled electro-mechanical analysis and application for typical components in MEMS,” Journal of Mechanical Strength, Vol. 23, No. 4, pp. 503, 2001. 6. References [12] P. M. Osterberg, S. D. Senturia, “M-test: A test chip for MEMS material property measurement using electro- statically actuated test structures,” Journal of Microelec- tromechanical Systems, Vol. 6, No. 2, pp. 107, 1997. [1] H. Y. Ma, Y. H. Wang, M. L Wang, et al. “The sequential coupling analysis of a cantilever driving by electrostatic force in an optical switch,” Optical Instruments, Vol. 25, No. 3, pp. 17, 2003. |