Fluid-Solid Interaction and Modal Analysis of a Miniaturized Piezoresistive Pressure Sensor

Dimitrios M. Tsamados Institut de Microélectronique, Electromagnétisme et Photonique (IMEP),

23 rue des Martyrs, B.P.257, 38016 Grenoble CEDEX 1, FRANCE Abstract

In the field of MEMS (MicroElectroMechanical System) devices fabrication as for every other field of microelectronics, numerical multiphysics simulation of the electric (and in our case mechanical) behavior of the devices is needed prior to fabrication in order to reduce research and production costs. In the present paper an ANSYS Multiphysics finite element analysis of a miniaturized pressure sensor based on Silicon On Insulator (SOI) technology is presented [ref. 1,2]. Fluid-Solid Interaction algorithms have been used, in order to simulate the sensor's response with the presence of air. The pressure sensor type is a piezoresistive one with special vent channels for having access to the reference pressure. The particularity of this kind of sensor design is that the gauges are made of monocrystalline silicon and are incorporated into the sensor's membrane. That means that the precise calculation of the stresses throughout the membrane's volume and consequently on the gauges is of great importance for estimating their final sensitivity. Stresses all over the gauges' volume are summed and multiplied by the piezoresistive coefficients of silicon in order to calculate the relative change of resistance. Optimization of the gauges' dimensions and placement has been performed having as a goal their maximum sensitivity. The position of the sensor's vent channels has been also studied and optimized to minimize its impact on the gauges sensitivity. Modal analysis of the sensor is also performed in order to determine the different modes of resonance and their corresponding frequency. The results of the analysis are then used to design the masks for the sensors' fabrication using typical microelectronics CAD tools.

Introduction MEMS devices are constantly gaining momentum in many fields of applications where sensors or actuators need to be integrated in microelectronics circuits. The applications are countless and they cover multiple fields of physics. Microfluidics and active control of flow is one of those fields that present great interest mostly for avionics. The pressure sensor simulated with ANSYS Multiphysics 7.0 package consists of a thin rectangular (typically 2µm or less) two-layer (Si and SiO2) membrane. Underneath this membrane (typical size 100x100µm2) there is a cavity of about 40-50µm that is connected to the reference pressure environment with specific vent channels. Standard photolithography and ion-implantation methods used in microelectronic circuit fabrication have been implemented to pattern the piezoresistive gauges in the sensor's membrane. When an external pressure is applied on the membrane the latter is deformed and stresses appear throughout the entire volume of the membrane. Silicon is a piezoresistive material and accordingly when stresses appear in the material's volume its resistivity changes. The sign of this variation depends on the direction of the stresses in relation to the crystallographic axes and the current's orientation in the gauge [ref. 3]. A typical electrical configuration of the sensor's gauges is a Wheatstone bridge [ref. 4]. With respect to materials, only the stiffness matrix and the piezoresistive coefficients of silicon are needed for a simple structural analysis. In the case of FSI analysis the properties of air and eventually those of other fluids like water are needed in order to obtain a solution. Residual stresses in the different layers of silicon or silicon dioxide, which are tightly related to the technological procedure, are not yet introduced due to lack of data regarding these parameters.

Full electro-mechanical analysis was not required in order to obtain the gauges' sensitivity and response to various pressure levels. As it is explained later in this paper, the calculation of the sensitivity of the gauges is quite straightforward and it only demands the knowledge of the stress distribution in the gauges and the corresponding piezoresistive coefficients, so no coupled-field electromechanical simulation of the Wheatstone bridge is required. Finally modal analysis using Block-Lanczos algorithm has been performed to determine the resonant modes of the structure in vacuum.

Procedure

The sensor's 3D model First of all, in order to save computational resources (memory, cpu time and hard disk space) the sensor's symmetry has been exploited (figure 1). As a result only the quarter of the sensor has been modeled and meshed. One of the greatest problems in MEMS devices FEA is the high aspect ratio of the structures, which may be of critical importance regarding the analysis validity (in our case the physical aspect ratio "thickness/length" of the membrane is of the order of 100-200 which is quite high for FEA). At the same time we have tried to avoid degenerate elements like pyramids or tetrahedra in order to have the maximum accuracy in our results [ref.5]. Consequently, the only possible solution to this problem regarding our pressure sensor is to have the finest possible meshing of the 3D model using hexahedral elements. Great care has been taken of the meshing of the sensor's membrane and particularly at the edges where we expect the highest gradients of stress.

Figure 1. 3D model of the 1/4 sector of the pressure sensor

In order to facilitate the optimization of the sensor's design, the structure's dimensions, meshing constants and material properties are introduced as parameters at the beginning of the analysis by using dialog boxes (*ASK command). This ameliorates the conviviality of the analysis and permits the use of the program by

non-ANSYS-specialized users, a factor of great importance in laboratory environment. All physical quantities (constants, dimensions, material properties) are introduced by strictly using the International System (SI). The latter is very important for the control and post-processing checking of the analysis and eventually for the debugging of the code. ANSYS modeler has been used for the construction of the 3D model of the sensor (figure 2). The "starting material" was a block, which then has been divided to individual volumes by using Boolean functions like "Divide Volume by Working Plane" (VSBW). Due to the structure's high aspect ratio, a BTOL command has to be used in order to determine the tolerance for these operations (typical value equal to 10-7). The volumes of the fluids and those of solids were created at the same time. Special care has been taken to avoid sharing of common areas between fluid and solid volumes at their interfaces due to the type of FSI analysis used. The "Sequential Weak Coupling Analysis" for FSI requires that fluid and solid volumes do not share the same areas in order to correctly assign the fluid-solid interface flags. A second model has also been created without the vent channels in order to investigate their impact on the membrane's deformation and stresses. The vent channels of the first model were 5µm wide and their depth has been set equal to the cavity's depth (40-50µm) for simplicity, although in reality the depth is smaller due to the etching method used (Deep Reactive Ion Etching, DRIE [ref. 6]) whose etching speed depends on the aspect ratio of the feature being etched (aspect-ratio-dependent etching or ARDE, [ref. 6]). For the FSI analysis the "sequential weak coupling analysis" has been used. As a result, first the fluid volumes had to be meshed and then the solid volumes. Fluid-solid interaction interfaces were flagged by issuing the FSIN command. The elements SOLID95 and FLUID142 (KEYOPT(4) activated) have been used to mesh the solid and fluid regions respectively (figures 3 and 4). Structural-only analysis has been also performed in order to compare the results with those of the FSI analysis (deformation, stresses) and eventually perform sub-modeling of the regions of interest (the gauges for instance). In that case FLUID142 element has been replaced by the null element MESH200 and all the FLOTRAN options were deactivated. Also in the latter case both SOLID95 and the anisotropic structural SOLID64 elements have been used interchangeably (only for silicon) in order to check if the silicon anisotropy regarding its stiffness matrix was critical for the final results. This element has not been used in the FSI analysis because in the ANSYS documentation it is not mentioned among the structural elements compatible with fluid elements for an FSI analysis [ref. 7]. Of course, comparisons between FSI and structural-only analyses were possible only when the SOLID95 element was used. As it is mentioned above, the critical regions of the pressure sensor are the membrane and its edges. Both two layers of the membrane were divided in 3 sub-layers (for the 1µm case) each one in order to have better stress gradient throughout their thickness (figure 5). It is obvious that more divisions could produce better results but the hardware limitations had to be taken under consideration also. Higher number of divisions in those layers would have meant that the divisions on the X and Y axes should also diminish to avoid high aspect ratio elements that could render the calculations impossible to carry out even on a relatively powerful workstation (HP Visualize j6000 with 4GB of RAM running HP-UX 11). Even with this kind of meshing (3 divisions) in some cases warning massages for the aspect ratio of some elements appeared but the number of these elements was small and, after checking ("Check Mesh"), they did not influence the final results. That kind of meshing problems is very common in MicroElectroMechanical Systems finite element analysis, especially when large-area suspended thin-film structures have to be modeled.

Figure 2. Pressure sensor's 3D Model

Figure 3. Solid region meshing

Figure 4. Fluid region meshing

Figure 5. Membrane meshing detail

Analysis

Boundary conditions Beginning from the FSI model, boundary conditions for both the solid and fluid regions of the model had to be introduced. First of all, symmetry boundary conditions were necessary because, as it is mentioned earlier, only the quarter of the whole structure has been modeled. As the fluid in the sensor's cavity changes shape while the overlying membrane is deformed the use of generalized symmetry boundary conditions for the fluid is obligatory [ref. 8]. The fluid symmetry conditions have been applied on the fluid area lying on these boundaries by issuing the command DA,ALL,ENDS,-1,0. Typical symmetry boundary conditions have been applied to the solid elements. Pressure boundary conditions (constant value = 0) were also necessary at the outlets of the vent channels so that the solution could converge. Key issue in the FSI analysis was the flagging of the fluid-solid interfaces. Displacement DOFs were also set to zero on all the nodes of the exterior faces of the pressure sensor except those situated on the sensor's upper surface and on the symmetry faces of course. A sinusoidal pressure signal has been applied on the sensor's membrane in the case of FSI analysis (transient) and a pressure of constant value in the case of purely structural analysis. For modal analysis only the structural symmetry and constrained DOFs have been applied as for the previous analyses and no acoustic elements (FLUID30 and FLUID130) were present.

FSI Analysis For the FSI analysis air (AIR-SI) was used as the fluid in the pressure sensor's cavities and vent channels. Laminar incompressible flow option has been used for the FLOTRAN analysis (low Reynolds number) and the mesh morphing option (Arbitrary Langrangian Eulerian formulation) for the fluid has been activated (FLDATA1,SOLU,ALE,1). Conservative interpolation option has been used for the transfer of loads between fluid and solid regions. Resolution of the structural equations has initiated the analysis and then the fluid equations were resolved by using the results of the first solution as boundary conditions for the FLOTRAN analysis, as the "Sequential Weak Coupling Analysis" requires. Geometrical non-linearities were also activated for the solid regions of the model (NLGEOM,ON). This option was mostly necessary for models with thin membranes, but it was maintained activated for all the models. As it was mentioned before, the FSI analysis was a transient one in order to observe the influence of air to the membrane's deformation and movement and compare the results with those produced by structural-only analysis. Eventually, other fluids could be used if the research is oriented towards a microfluidics application by inversing the sensor's function and use it as a thermally or electrostatically actuated pump. Another option, which is currently under testing, is the integration of the sensor in a microfluidics channel where a fluid like water or ethanol flows through it, in order to measure the pressure applied to the channel's walls at a steady state (constant fluid velocity). Further results on these two options were unavailable during the writing of this paper. During the early analyses a very low frequency pressure signal (1s period and 105 Pa amplitude=>∆P=105Pa because P=0 at the vent channels' outlets) was applied on the pressure sensor's membrane to guarantee good convergence as the fluid-solid relaxation factor has been set equal to 1 (FSRE). This analysis is adequate in order to compare the results (stresses, deformations) when air is present in the cavity with those where there is no fluid present. In addition, hardware limitations become significant if smaller period signals are used because more iterations are needed for the solution to converge. On the other hand, diminishing of the meshing density in order to "lighten" the model could result to not very accurate results.

Structural-only Analysis During this analysis constant pressure has been applied on the sensor's membrane. As in FSI analysis, the NLGEOM option was active. The pressure range was between 102 to 106 Pa. The "program chosen" option was activated as for the type of solver and maximum number of iterations was set to 15. In all cases

convergence was fast and the comparison with the results of simulations with NLGEOM deactivated showed that for low-pressure levels the option was not needed as it was anticipated by the plate theory [ref. 9]. For more accurate results, submodeling was an option. In ANSYS documentation [ref.10] we can find the following definition of submodeling: "Submodeling is a finite element technique used to get more accurate results in a region of your model. Often in finite element analysis, the finite element mesh may be too coarse to produce satisfactory results in a region of interest, such as a stress concentration region in a stress analysis. The results away from this region, however, may be adequate.To obtain more accurate results in such a region, you have two options: (a) reanalyze the entire model with greater mesh refinement, or (b) generate an independent, more finely meshed model of only the region of interest and analyze it. Obviously, option (a) can be time-consuming and costly (depending on the size of the overall model). Option (b) is the submodeling technique.". In our case submodel had to be carefully chosen in order to include large part of the substrate so that the stress gradients in the gauges were correct. That means that the number of equations to resolve remained relatively high due to refinement of the meshing in that model. Consequently, in terms of calculation time it is possible to obtain times equal to or even higher than the coarse model solution ones. On the other hand, we only have to run the coarse model solution once, save it and import the results as boundary conditions to the different submodels. Finally, if the submodels are reasonably meshed (in our case the smallest element size was 0.1µm of thickness and the maximum aspect ratio = 10 for maximum structure dimensions of 40x5x1µm3) the ratio "calculation time over accuracy obtained" can be quite satisfying. For the optimization of the gauges positioning as a function of their sensitivity the "optimization" algorithm available in ANSYS has been utilized. If a Wheatstone bridge is used there are two types of gauges in the sensor's membrane, those parallel to the membrane edge and those perpendicular to them (figure 6)[ref. 4]. The sensitivity of the gauges is calculated by summing the element stress results in each gauge's volume for each direction separately (X, Y and Z). In all cases, if a Wheatstone bridge configuration is used (constant

voltage supply), the pressure sensitivity can be expressed by the ratio ∆RR

1∆P

, in other words only the

relative change of the gauge's resistance is important for a given differential pressure ∆P [ref. 4]. This ratio is directly related to the product of piezoresistive coefficient times the stress for each direction following the formulation: ∆RR

=σ lπ l +σ tπ t

where the indices t and l concern the orientation of the stresses with respect to the current flow direction and σ and π are the corresponding stresses and piezoresistance coefficients respectively [ref. 4]. The model was created using the X axis as the equivalent of (110) direction of the silicon crystal (figure 7). As a result

the piezoresistance coefficient πl ( π l =12π 11 + π 12 + π 44( ), [ref. 3]) will be multiplied by the stresses in the

X direction and πt ( π t =12π 11 + π 12 − π 44( ), [ref. 4]) will multiply the stresses in the Y and Z directions.

Typical values for the coefficients π11, π12 and π44 for p-type silicon have been used, that is [ref. 3]: π11= +6.6x10-11 Pa-1 π12= -1.1x10-11 Pa-1 π44= +138.1x10-11 Pa-1

Figure 6. Gauges in the membrane

Figure 7. Directions in the crystal

In addition, when the anisotropic element SOLID64 is used, the stress-strain matrix has to be introduced either in stiffness form ({D}) or in the flexibility form ({D-1}). Usually in the literature we find the stiffness coefficients of silicon with respect to the crystallographic orientations of the cubic system, as it was the case for the piezoresistance coefficients above. This means that in order to use an anisotropic structural element with the orientation as described before (X axis=>(110)) the stiffness matrix has to be adapted to the ANSYS coordinate system. The values for the stiffness coefficients for silicon in the cubic system can be found in the literature [ref. 11] and they are the following: D11=1.6564x1011 Pa D12=0.6394x1011 Pa D44=0.7951x1011Pa Following the conversion method of Wortman et al. [ref. 12] the stiffness matrix was finally adapted to the ANSYS c.s. In the case of the isotropic element SOLID95 the Young modulus has been set equal to

165GPa and the Poisson's ratio v=0.22 for the silicon for all types of analyses (FSI, structural-only and modal). For this kind of analysis the second model without vent channels has also been simulated in order to optimize the vent channels' positioning and to study their influence on the sensor's sensitivity by comparing the results with those of the complete model.

Modal Analysis For the modal analysis the method "Block Lanczos" has been used and 10 modes have been expanded with element calculation for all of them. For the first simulations no damping factors have been introduced. Acoustic elements will be used in the near future to better simulate the damping effects taking place into the structure due to the presence of air. These results may be available for the ANSYS Conference in May. Both SOLID95 and SOLID64 elements have been used for this analysis in order to compare the results. All material properties were the same as in the previous analyses. Additionally, the second model without vent channels has also been simulated for comparison.

Analysis Results & Discussion

FSI Analysis and Structural-only Results and Comparison The most important results of this analysis are the minimum and maximum values of stresses in the structure, their distribution and of course the maximum deformation of the membrane. Also the fluid velocity, pressure and flow in the sensor's cavities are studied. The structural results are then compared to those of the "Structural-only" analysis. The comparison has shown that when air is present in the cavity (40µm deep) and the vent outlets are not blocked so that the air is free to flow, the difference in the membrane's deformation and in the stresses throughout its volume are negligible compared to the structural-only analysis. We have to underline once more that the pressure signal in the FSI analysis had an amplitude of 105Pa (as in the structural-only) and a period of 1 second. The maximum velocity of air at the outlets of the vent channels was of the order of 1µm/s as the fluid is forced to flow through the relatively narrow shafts (figures 8 and 9).

Figure 8. Vector plot

Figure 9. Flow trace

The positioning of the vent channels was of great importance and the simulations have given very useful information about this parameter. The stresses are higher towards the center of each edge of the membrane than at its corners. As it has been explained in the "analysis" section, the sensor's sensitivity depends highly on the level of stresses on each gauge as the membrane is deformed. The simulations showed that if the vent channels are originating from the sensor's central lateral part then the level of stresses diminishes and so does the sensitivity. On the other hand, if the channels' origins are placed at the corners of the sensor where the stresses are low anyway and the gauges are not likely to be placed there than the impact of these channels on the sensor's sensitivity becomes negligible (figure 10).

Figure 10. Von Mises stresses

Finally, the most important part of the simulations was the optimization of the gauges dimensions and positioning as well as the membrane's thickness and surface. The difficult and time-consuming part of this optimization was that it had to be performed for a wide range of pressures. As one can easily understand the post-simulation qualitative analysis and physical interpretation of the results is quite long. Due to the large number of parameters that had to be studied separately the one from the other we will try to give in the small space of this article some general tendencies resulting from the analyses as well as some quantitative results concerning major parameters such as the gauges dimensions and the membrane thickness. As we can observe in figure 11 the gauges' absolute value of ∆R/R ratio increases while the membrane thickness decreases. The same thing happens to the gauges parallel to the membrane's edge when their width decreases (figure 12). By diminishing the perpendicular gauges' length we can augment their ∆R/R ratio (figure 13). In general, and particularly for the gauges parallel to the membranes edges, when their distance from the edge increases their sensitivity is decreasing quite rapidly due to the high stress gradient near the edges. This result is in agreement with already existing results published elsewhere [ref. 4]. Very slight differences in stress levels are observed when the anisotropic element SOLID64 (slightly higher stresses) was used in the place of the SOLID95 but the general trends have not been altered due to this change.

Figure 11. ∆R/R function of membrane thickness

Figure 12. ∆R/R function of gauge's width

Figure 13. ∆R/R function of gauge's length

Modal Analysis The resonant modes of the pressure sensor have been extracted during this analysis for the two different designs of the sensor, the one with vent channels and the other without. No significant differences have been observed when vent channels are present either if they are placed at the corners or at the center of the pressure sensor's cavity. For the 100x100x2µm3 membrane the fundamental mode was situated at a frequency of approximately 2.19MHz. In the table below the frequencies of the first 10 modes are given. Frequencies of the first 10 modes of resonance SET TIME/FREQ LOAD STEP SUBSTEP CUMULATIVE 1 0.21940E+07 1 1 1 2 0.80243E+07 1 2 2 3 0.80731E+07 1 3 3 4 0.13350E+08 1 4 4 5 0.18536E+08 1 5 5 6 0.18904E+08 1 6 6 7 0.19016E+08 1 7 7 8 0.19473E+08 1 8 8 9 0.21695E+08 1 9 9 10 0.23831E+08 1 10 10

Conclusion In this paper the fluid-solid interaction, purely structural and modal analyses of a miniaturized Silicon On Insulator technology pressure sensor are presented. Details concerning the parametric 3D model and its meshing are analyzed and explained along with the methodology followed for obtaining the final solution

and the results. The design optimization algorithm offered by ANSYS has been utilized in order to come up with the highest sensitivity layout of the gauges on the sensor's membrane. Results of this optimization have also been presented. Comparison between the results of the FSI analysis and the structural-only has been effectuated. In the near future derivative designs of the presented pressure sensor will be simulated for applications in microfluidics. In overall, ANSYS Multiphysics package turned out to be well suited for the numerical simulation needs regarding this pressure sensor.

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