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| ARTICLE |
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| Year : 2011 | Volume
: 57
| Issue : 2 | Page : 118-124 |
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A Modified Method to Model the Effect of Residual Stress on Touch-point Pressure and Pull-in Voltage for MEMS Capacitive Transducers using Square Diaphragm
Anurekha Sharma
Department of Electronic Science, Kurukshetra University, Kurukshetra, Haryana - 136119, India
| Date of Web Publication | 30-May-2011 |
Correspondence Address:
Anurekha Sharma
Department of Electronic Science, Kurukshetra University, Kurukshetra, Haryana - 136119
India
 DOI: 10.4103/0377-2063.81739
 Abstract | | |
The silicon-based materials used for fabricating the square diaphragm for Micro-electro-mechanical systems capacitive transducer have inherent stress. This stress is known as residual stress. The presence of stress influences the mechanical behavior of the material, thereby influencing the performance characteristics of the devices. A methodology discussed earlier for modeling the small deflection of square diaphragm has been extended to take into account the effect of residual stress on the pull-in voltage and touch-point pressure. The results are compared with simulated and experimental ones. Keywords: Diaphragm, Micro-electro-mechanical systems, Pull-in voltage, Residual stress, Touch-point pressure
How to cite this article:
Sharma A. A Modified Method to Model the Effect of Residual Stress on Touch-point Pressure and Pull-in Voltage for MEMS Capacitive Transducers using Square Diaphragm. IETE J Res 2011;57:118-24 |
How to cite this URL:
Sharma A. A Modified Method to Model the Effect of Residual Stress on Touch-point Pressure and Pull-in Voltage for MEMS Capacitive Transducers using Square Diaphragm. IETE J Res [serial online] 2011 [cited 2013 May 22];57:118-24. Available from: http://www.jr.ietejournals.org/text.asp?2011/57/2/118/81739 |
| 1. Introduction | |  |
An MEMS capacitive transducer is a sensor or an actuator that employs a parallel plate structure and facilitates monitoring of capacitance change due to an external mechanical excitation, such as force, acoustical pressure or acceleration [1] or electrostatic excitation in case of actuators. The parallel plates comprise a deformable electrode and a fixed electrode with an intervening layer of dielectric over it in transducers to allow touch-mode operation or to avoid electric short in electrostatic actuators at pull-in as shown in [Figure 1]. The deformable electrode is usually a clamped diaphragm and can be fabricated using different materials and different geometries such as, circular, square and rectangular. Square diaphragms are used in numerous MEMS structures because of better area efficiency and process capability using IC lithography [2] . Besides touch-mode capacitive pressure sensors [2] , square diaphragms find use in numerous applications like electrostatic valve actuator for high-pressure applications [3] , polysilicon micromirrors [4] , silicon capacitive microphone [5] , micropumps [6] and bio-medical applications [7] . Different materials used for fabrication of diaphragm are boron doped silicon [8] , polysilicon, silicon nitride [9] and polyimide [10] . All these materials are known to have residual stresses. The residual stress affects the device behavior by influencing its touch-point pressure and pull-in voltage. The pull-in voltages of micro test structures can be used to extract the material parameters of thin films, such as Young's moduli and residual stresses [11],[12] . Determination of the pull-in voltage is critical in the design to determine the sensitivity, instability in the operational range and the dynamics of devices. Accurate determination of the pull-in voltage is very challenging by virtue of the mechanical-electrical coupling effect and the nonlinearity of electrostatic force. Several methods like Finite Element Method (FEM), lumped model approach and solving coupled partial differential equations using numerical techniques are available to find the pull-in voltage [13] . Simple fast solutions are available for determination of pull-in voltage of cantilever beams, fixed-fixed beams and circular diaphragms with excellent accuracies and can determine the pull-in voltage for the mentioned structures within 1% agreement with FEM results under certain limitations [14] . However, published analytical or empirical solutions to determine the pull-in voltage for square diaphragm predict pull-in voltages that show significant error when compared with the finite element analysis results or experimentally measured values [14] . Analytical model [14] based on a linearized uniform approximation model of the electrostatic pressure and a 2-D load deflection model under uniform pressure gives the expression of pull-in voltage by assuming that the pull-in occurs at a critical displacement equal to one-third of the gap between the electrodes. A method has been proposed earlier [15] by the same authors to solve the fourth-order partial differential equation by using a trial solution for small deflection of plates and the same is extended to take into account the effect of residual stress on pull-in voltage and touch-point pressure. The closed form expression of pull-in voltage and critical distance are the outcome of the solution. The variation of capacitance with voltage is also obtained. Another distinct feature of the method is that the deflection versus pressure graph depicts a realistic situation as no further deflection takes place after touch-point pressure is reached.
| 2. Theory | | |
For the plates with residual stress, the governing equation is [16]
where w(x, y) is the deflection at any point (x, y) of the diaphragm, σ is the residual stress, h is the thickness of the diaphragm and P is the distributed pressure load. D, flexural rigidity, is given by
where E is the Young's modulus of the diaphragm material and υ is the Poisson's ratio. In the presence of applied pressure and applied voltage, Equation (1) is modified as
where Pel is the electrostatic pressure and P is the mechanical pressure. The electrostatic pressure Pel is given as
where d0 is the distance between the plates, εr is relative permittivity of the medium or the dielectric constant of the medium and ε0 is permittivity of free space. The diaphragm deflection w(x, y) with air as dielectric (dielectric constant εa ) is given by considering that the distance d0 between the plates changes to (d0 - w) due to the displacement w of the diaphragm in the presence of applied pressure and voltage as shown in [Figure 1]. Substituting Equation (3) in Equation (2) for diaphragm without the intervening layer of dielectric, the equation becomes
For diaphragm with intervening layer of dielectric,
where εi is the dielectric constant of the insulator and tm is the thickness of the insulator. Equations (4) and (5) can be written in a generalized form as (with εa = 1)
where deff = tm /εi (for a single layer of dielectric) and deff = 0 for air.
To know the behavior of diaphragm with residual stress under the influence of applied mechanical and electrostatic load, one needs to devise a methodology to solve Equation (6). The next section discusses the semi-analytical technique to solve these equations.
| 3. Semi-Analytical Technique | | |
This section discusses the methodology adopted for solving Equation (6) to achieve the closed form expression for pull-in voltage and touch-point pressure as a function of residual stress for small deflection of diaphragm. Following assumptions are made to simplify the calculations:
- small deflection regime is assumed that the maximum deflection does not exceed one-third of the thickness of the diaphragm [2] ;
- residual stress is assumed to be constant throughout the diaphragm thickness; and
- fringing field effect is ignored.
The boundary conditions for the square diaphragm with clamped edges are as follows:
The deflection is
where 2a is the side length of the diaphragm [Figure 2].
The trial solution that satisfies the above given boundary conditions is
where λ is a function which depends on P, V, h, E, υ, σ and εr . It is evident that λ gives the maximum deflection at a point x = 0 and y = 0. The deflection at any other point (x, y) can be calculated by knowing λ and substituting the value for x and y in Equation (7), for a given diaphragm of side length 2a. The value of λ can be evaluated by applying the new method as already described in [15] by authors. The governing differential Equation (6) is minimized and it is assumed that w(x, y) is orthogonal with respect to all other co-ordinate functions [17] , i.e.
The above equation has no closed form analytical solution. It can be solved numerically or by linearizing the electrostatic pressure by using Taylor series [18] . For obtaining analytically closed form solutions, it is necessary to consider the modification in the applied electrostatic load as a result of change in gap due to deflection. To evaluate this, numerically successive iterations are to be carried out till the resultant defection reaches the equilibrium value. However, for obtaining the analytical solution, one starts by obtaining the initial value of λ, i.e. the starting value of deflection as soon as the voltage is applied. Initially, when voltage is applied, the value of λ from Equation (8) is found by putting w = 0 and solving Equation (8) to obtain
This is the starting value of λ, which gives starting deflection as soon as the voltage is applied. However, this is not steady-state value of λ. The steady-state value is obtaining by modifying the deflection at the center in view of the positive feedback because of change in electrostatic pressure brought about by decrease in gap. This is done by substituting Equation (9) in Equation (7) to get the initial deflection at any point (x, y) as
However, with this deflection, the gap reduces to d0 - w(x, y) and one has to calculate the new value of deflection again. Since the trial solution governing deflections remains the same as given by Equation (7) except for the numerical value of λ, without any loss of generality, one can write
On substituting Equation (7) for w(x, y) and taking x = 0 and y = 0, for maximum deflection, one obtains the following equation in λ
Equation (11) has got three roots and only one of them gives a stable value. The value of λ is substituted in Equation (7) to get the deflection at any point (x, y) in terms of applied voltage, pressure and stress.
3.1 Pull-in Voltage
Pull-in voltage is determined by differentiating the deflection w(x, y) at the center, i.e. at x = 0, y = 0 with respect to voltage in the absence of any mechanical pressure and equating δV/δw to zero. The solution of this differential gives the critical distance. The pull-in voltage is calculated by substituting the wcr for w(x, y) in the expression for deflection. The closed form expression for pull-in voltage (Vpull ) is
Critical distance w cr is
[Table 1] gives the comparison of the pull-in voltage obtained from the model with experimental values reported by other researchers. It can be seen from the table that in the case of small deflections, i.e. when thickness of the diaphragm is greater that the air gap, the calculated values are closer to the reported ones. However, in large deflection case there is considerable deviation from the reported values of pull-in. For a diaphragm [19] with a = 1.0 mm, h = 1.1 μm, d0 = 3.75 μm, ε0 = 8.85 × 10−12 F/m, E = 170 GPa, ν = 0.3, σ = 65 MPa, [Figure 3]a compares the deflection variation with voltage with that obtained from simulation and [Figure 3]b shows the results obtained from Intellisuite® . The calculated pull-in is 21.04 V, whereas simulated pull-in is 23.9 V. | Figure 3: (a) Center deflection with voltage for a diaphragm with a = 1.0 mm, h = 1.1 µm, d 0 = 3.75 µm, ε 0 = 8.85 × 10−12 F/m, E = 170 GPa, ν = 0.3, σ = 65 MPa; (b) center deflection with voltage for a diaphragm with a = 1.0 mm, h = 1.1 µm, d 0 = 3.75 µm, ε 0 = 8.85 × 10−12 F/m , E = 170 GPa, ν = 0.3, σ = 65 MPa obtained from Intellisuite®.
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3.2 Capacitance Computation
The amount of displacement brought about by application of pressure or voltage is measured in terms of change in capacitance. The capacitance for the capacitive transducer with air as dielectric is given as:
The capacitance is obtained numerically by dividing the entire diaphragm into small elements of the size of 0.1a × 0.1a, starting from co-ordinate (−a, −a) to (a, a). The deflection of each element is calculated by substituting the value of λ obtained from Equation (11) in Equation (7) and taking (x, y) as (-a + Δa, -a + Δa), (-a + 2Δa, -a + 2Δa), (-a + 3Δa, -a + 3Δa)--------------(+a, +a), where Δa is 0.1a. [Figure 4]a compares the capacitance obtained from calculation with that obtained from FEM simulation using Intellisuite® . [Figure 4]b gives the simulated results for the above diaphragm obtained from Intellisuite® . | Figure 4: (a) Capacitance versus voltage for a diaphragm with a = 1.0 mm, h = 1.1 µm, d 0 = 3.75 µm, ε 0 = 8.85 × 10−12 F/m, E = 170 GPa, ν = 0.3, σ = 65 MPa; (b) capacitance versus voltage for a diaphragm with a = 1.0 mm, h = 1.1 µm, d 0 = 3.75 µm, ε 0 = 8.85 × 10−12 F/m, E = 170 GPa, ν = 0.3, σ = 65 MPa obtained from Intellisuite®.
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3.3 Touch-point Pressure
Touch-point is defined as the pressure at which diaphragm just touches the fixed electrode and is of importance for the design of touch-mode capacitive pressure sensors [2] . The touch-point pressure as well as the deflection variation with pressure depends only on gap d0 and not on the presence or absence of dielectric. Therefore, for calculation of touch-point pressure as well as for variation of deflection with pressure, deff is zero, as the dielectric need not be considered. As soon as the diaphragm makes contact with the underlying dielectric, the deflection w(0, 0) becomes constant. The touch-point pressure (Ptouch ) is found out by differentiating the deflection w(x, y) at the center, i.e. at x = 0, y = 0 with respect to pressure P at zero voltage and equating δw/δP to zero, as after touch-point is reached, there is no further deflection in the vertical direction. The closed form expression for touch-point pressure is
[Table 2] compares the touch-point pressure obtained from this model with that obtained by simulation using Intellisuite® .
[Figure 5]a and b compares the deflection with applied pressure as obtained from the proposed model with those simulated using Intellisuite® for a square diaphragm with a = 250 μm, h = 20 μm, d0 = 8 μm, ε0 = 8.85 × 10−12 F/m, E = 130 GPa, deff = 0, ν = 0.3 with and without residual stress. [Figure 6] compares the deflection versus pressure characteristics of the diaphragm [19] with a = 1.0 mm, h = 1.1 μm, d0 = 3.75 μm, ε0 = 8.85 × 10−12 F/m, E = 170 GPa, ν = 0.3, σ = 65 MPa with the results obtained from another analytical technique [24] . It is evident that the present technique gives a realistic picture, with the deflection constrained by the gap. | Figure 5: (a) Center deflection versus pressure for a square diaphragm with a = 250 µm, h = 20 µm, d 0 = 8 µm, ε 0 = 8.85 × 10−12 F/m, E = 130 GPa, deff = 0, ν = 0.3 without residual stress; (b) deflection versus pressure for a square diaphragm with a = 250 µm, h = 20 µm, d 0 = 8 µm, ε 0 = 8.85 × 10−12 F/m, E = 130 GPa, deff = 0, ν = 0.3 with residual stress
σ = 50 MPa.
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 | Figure 6: Comparison between the present work and reported work for center deflection with pressure for a diaphragm with a = 1.0 mm, h = 1.1 µm, d 0 = 3.75 µm, ε 0 = 8.85 × 10−12 F/m, E = 170 GPa, ν = 0.3, σ = 65 MPa.
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| 4. Conclusion | | |
This paper deals with incorporating the effect of residual stress in determining the behavior of a single-layered square diaphragm. The closed form equation derived in this paper forms a good basis of extracting the residual stress, in case the deflections are determined on the basis of the pull-in phenomenon. Closed form solutions of pull-in voltage and touch-point pressure have been derived for the single-layer diaphragm using a simple and a new technique. The results are found to be in good agreement with the simulated as well as the reported results. The pull-in voltage and touch-point pressure increase with the tensile stress and decrease with compressive stress. The technique demonstrated is hence effective for computing the residual stress by substituting the measured data in the model. The closed form expressions allow finding the pull-in voltage and touch-point pressure of a capacitive transducer, once the value of residual stress has been measured. The advantage of the technique lies in its simplicity and reduced computation time as compared to finite element method, while still giving reasonably good accuracy. The technique has the distinct advantage of combining the effect of pressure and voltage on the deflection behavior of the diaphragm, and therefore can be applied to capacitive sensors as well as actuators employing square diaphragm with clamped edges. The change in capacitance is used to sense the applied pressure and voltage. Therefore, the computation of capacitance is very important. The model provides for the computation of capacitance with pressure as well as voltage. The load-deflection characteristics obtained for applied pressure are restrained by the gap between the electrodes, i.e. once the touch-point is reached, further application of the pressure results in increase in contact area without any further deflection.
| 5. Acknowledgments | | |
The author acknowledges the support of CEERI for Intellisuite® . Acknowledgments are due to Mr. Jaideep Gupta, Lecturer, NIT, Kurukshetra, for ANSYS® and Mathematica® .
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| Authors | | |
 Anurekha Sharma is working as Associate Professor in Department of Electronic Science, Kurukshetra University, Kurukshetra. She obtained M.Tech in Electronics, Computer and Communication Engineering from NIT Kurukshetra and Ph.D from Kurushetra University. Her field of interest as well as research is MEMS modeling and design. She has published about a dozen of research papers in the field of MEMS in International Journals.
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6]
[Table 1], [Table 2]
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