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A Model for Predicting the Piezoresistive Effect in Microflexures Experiencing Bending and Tension Loads Brigham Young University BYU ScholarsArchive Faculty Publications 2008-02-07 A Model for Predicting the Piezoresistive Effect in Microflexures Experiencing Bending and Tension Loads Gary K. Johns [email protected] Larry L. Howell [email protected] Brian D. Jensen Timothy W. McLain [email protected] Follow this and additional works at: https://scholarsarchive.byu.edu/facpub Part of the Mechanical Engineering Commons BYU ScholarsArchive Citation Johns, Gary K.; Howell, Larry L.; Jensen, Brian D.; and McLain, Timothy W., "A Model for Predicting the Piezoresistive Effect in Microflexures Experiencing Bending and Tension Loads" (2008). Faculty Publications. 202. https://scholarsarchive.byu.edu/facpub/202 This Peer-Reviewed Article is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Faculty Publications by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. 226 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2008 A Model for Predicting the Piezoresistive Effect in Microflexures Experiencing Bending and Tension Loads Gary K. Johns, Larry L. Howell, Fellow, ASME, Brian D. Jensen, Member, ASME,and Timothy W. McLain, Senior Member, IEEE Abstract—This paper proposes a model for predicting the or a membrane). In the proposed integrated piezoresistive self- piezoresistive effect in microflexures experiencing bending sensing devices, however, rather than the piezoresistive element stresses. Linear models have long existed for describing piezore- being part of the flexure, it is the flexure. The loading condi- sistivity for members in pure tension and compression. However, extensions of linear models to more complex loading conditions tions for such sensors are more complex than pure tension or do not match with experimental results. A second-order model compression. to predict piezoresistive effects in tension, compression, and more This paper shows that analytical resistance data from the complex loading conditions is proposed. A reduced form of the current model of piezoresistivity do not match with the ex- general second-order model is presented for thin flexures in bend- perimental data for microflexures in bending, and introduces ing. A three-step approach is used to determine the piezoresistive coefficients for this reduced-form model. The approach is demon- a second-order model to predict the piezoresistive effect under strated for two sets of n-type polysilicon. The predictive ability of more complex loading conditions. This model is demonstrated the model is investigated by comparing the results to the experi- by determining the piezoresistive coefficients for n-type poly- mental results using the new piezoresistive model and coefficients. crystalline silicon. The model and resulting piezoresistive co- One of the ways to implement the model is with multiphysics efficients are used in a case study to model the piezoresistive finite element analysis (FEA). The piezoresistive FEA for flexures algorithm is a FEA implementation of the unidirectional form of behavior of two microdevices. the model for flexures. The results presented in this paper are for the simplified cases of long thin flexures experiencing bending and axial loads. This new model could contribute to optimized II. BACKGROUND sensors and feedback control of microdevices, nanopositioning, and self-sensing microdevices. [2007-0052] Piezoresistivity is a material property that couples mechan- Index Terms—Finite element methods, microelectromechani- ical strain to bulk electrical resistivity [5]. In other words, if cal devices, microsensors, modeling, piezoresistance, piezoresistive a piezoresistive member is bent, stretched, or compressed, its devices. electrical resistance changes. The change in resistance can then be measured or converted into a change in voltage, and the cor- responding stress (or strain) can be determined. Thus, stresses I. INTRODUCTION and strains can be related to a physical phenomenon such as NTEGRATED piezoresistive self-sensing is a sensing displacement or applied force through piezoresistivity [6]. I method that utilizes the deflection of intrinsic microflexures Currently, the piezoresistive effect is most widely used in to sense the state of a microdevice [1]–[4]. These microflexures various types of sensors. Sensors utilizing the piezoresistive deform under a force or displacement caused by a physical effect include microphones, pressure sensors, force sensors, phenomenon. Due to their piezoresistive material property, flow sensors, and displacement sensors [7], [8]. One of the most the electrical resistance of a flexure changes in correlation common techniques for fabricating piezoresistive sensors is to with its deformation. This change in resistance can be used diffuse a resistor onto a silicon diaphragm or cantilever beam to calculate the value of the original force or displacement. that deflects in reaction to an external stimulus [9], [10]. The Piezoresistive elements have often been implemented in mi- diffused silicon resistor is put into tension or compression as the crosystems in ways that place the piezoresistive elements in diaphragm or cantilever deflects. The resistance elements that pure tension or compression (such as on the surface of a beam formed from the diffused silicon can be electrically connected in a Wheatstone bridge configuration. The change in resistance Manuscript received February 28, 2007; revised October 3, 2007. Subject is then measured directly from a change in voltage. Editor H. Fujita. Many microdevices are compliant mechanisms [11], which G. K. Johns was with the Department of Mechanical Engineering, Brigham gain some of their mobility from deflection rather than pin Young University, Provo, UT 84602 USA. He is now with Merit Sensor Systems, Inc., South Jordan, UT 84095 USA. joints or other hinges. Compliant devices are particularly ad- L. L. Howell, B. D. Jensen, and T. W. McLain are with the Department of vantageous at the microlevel because they can be fabricated in Mechanical Engineering, Brigham Young University, Provo, UT 84602 USA. a single layer and require no assembly. Compliant mechanisms Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. also avoid the friction, wear, and backlash that are inherent in Digital Object Identifier 10.1109/JMEMS.2007.911874 joints with rubbing parts. As compliant beams (flexures) are 1057-7157/$25.00 © 2008 IEEE Authorized licensed use limited to: Brigham Young University. Downloaded on March 16, 2009 at 11:26 from IEEE Xplore. Restrictions apply. JOHNS et al.: MODEL FOR PREDICTING THE PIEZORESISTIVE EFFECT IN MICROFLEXURES 227 Fig. 1. Schematic of Charles Smith’s experimental setup. Coordinate axis 3 is into the page. deflected, their stress–strain state is changed, and thus, they are well suited for integrated piezoresistive sensors. Fig. 2. Analytical results and experimental data comparison of the super- A. Linear Model of Piezoresistivity position model [13]. The nominal resistance of the experimental device is 2266 Ω. In 1954, Charles Smith published a paper [12] exposing the piezoresistive properties of silicon and germanium. Experimen- other two orthogonal directions. By assuming that these lower tal data were gathered and used to derive the piezoresistive stresses are insignificant, (3) simplifies to coefficients that relate change in resistance to stress. The experi- ρ mental setup is shown in Fig. 1. A beam was put in tension, and Δ 1 = π11σ1. (4) the change in electrical resistance was measured. These data ρ were used to calculate the piezoresistive coefficient π11 as This model works well when modeling beams loaded in pure 1 ΔR tension or compression and for piezoresistive elements on one π11 = (1) surface of a membrane where they are in pure tension or S R0 compression. However, it will be shown that these models are where S denotes stress, and R is the electrical resistance. It was not adequate for more complex loading conditions. also found that silicon and germanium are much more sensitive to the piezoresistive effect than most metals. B. Superposition Model of Piezoresistivity in Bending The fractional change in resistivity is related to stress through the Π matrix, which is shown in a simplified form for the cubic Beams in bending simultaneously experience different levels crystal structure of silicon as of tension and compression throughout their cross sections and along their lengths. Smith’s linear model was considered as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ a possible method for predicting the piezoresistive effect in Δρ1 π11 π12 π12 000 σ1 beams undergoing bending because it has been shown to predict ⎢ Δρ2 ⎥ ⎢ π12 π11 π12 000⎥ ⎢ σ2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ the change in resistance of beams in tension or compression. A 1 ⎢ Δρ3 ⎥ ⎢ π12 π12 π11 000⎥ ⎢ σ3 ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ superposition form of the linear model was used to calculate ρ ⎢ Δρ4 ⎥ ⎢ 000π44 00⎥ ⎢ τ12 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ separately the change in resistivity for the tensile and compres- Δρ5 0000π44 0 τ13 sive segments of the beam. The principle of superposition was Δρ6 00000π44 τ23 (2) used to add these components together. To check the validity of the superposition model, resistance-deflection curves predicting where ρ is the original resistivity, and Δρi is the change in the change in resistance of beams under a bending load were th resistivity in the i direction. The terms σ1, σ2, and σ3 are the used. An n-type polysilicon device was used for
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