Load-Deflection Behavior of RF-MEMS Switches : FEA and Analytical Modeling for Prediction of Mechanical Properties †

SixNy/a-Si/SixNy thin film RF-MEMS switches were fabricated by unconventional PECVD process using surface micromachining approach. The mechanical properties of tri-layer were measured by nanoindentation and wafer curvature method. Deflections of switches clamped on two opposite edges were measured by a profilometer applying increasing quasi-point pressure loads. Finite Element Analysis (FEA) was used to study the mechanical behavior of clamped-clamped switches. An analytical solution was developed and validated, numerically and experimentally, to describe the load-deflection response of perforated membranes to quasi-point loads. The proposed function was used to determine the internal stress of the investigated membranes; the relative error between the predicted and calculated stress values was in the range 2.1–8.5%.


Introduction
Quality and control of advanced materials and processes are important for the fabrication of reliable and high-performance MEMS devices.Nowadays, predictive numerical models of the electro-mechanical behavior of the device allow for its optimized design, and more accurate results can be obtained if the mechanical properties of structural materials are known.The mechanical characterization of materials is usually performed on ad-hoc thin film test samples, however an evaluation by in-situ measurements performed on the final MEMS device could result more meaningful, reliable and useful.
A surface profiler can be used to measure the deflection under load of suspended structures (beams, cantilevers or membranes for example).Young's modulus and stress can be indirectly and approximatively determined from the load-deflection response, by using numerical methods of comparison or analytical solutions solving complex differential equations of load-displacement.In [1][2][3][4] the deflection of thin-and thick-films, cantilever, plates and shells under uniform loads and different conditions applied on the edges (simple support, clamping, free-edges) was discussed and correlate to the material mechanical properties.However, to date, to the best of our knowledge, there are not analytical solutions for the nonlinear large deflection of membranes with two opposite edges Clamped and the other two Free (CCFF), under quasi-point pressure loading.
This work propones a FEA and analytical modeling of nonlinear large deflection of clamped-clamped perforated membranes, for the extraction of mechanical properties from in-situ measurements performed by applying quasi-point loading with a stylus profiler.A numerical model of the load-deflection behavior of a CCFF perforated membrane was developed using Comsol Multiphysics software, in agreement with the measured deflection data.Multi-parametric FEA simulations allowed to develop an approximate analytical function of maximum displacement under quasi-point loading.The displacement was written as a function of the main geometric and mechanical parameters, hence Young's modulus and internal stress can be indirectly extracted from the deflection measurements.The proposed function was validated, numerically and experimentally; results demonstrated that the internal stress of the CCFF membranes convey the micromachining, with respect to the value measured on continuous thin films of the same structural tri-layer.

Materials and Methods
Micromachined membranes are used in many MEMS applications.In this work, unconventional 100 °C PECVD process was used for the fabrication of (190 nm)SixNy − (320 nm)a-Si − (210 nm)SixNy thin film membranes of different sizes and porosity.The membrane layer was patterned by standard photolithography and etched using IC standard TEGAL900 plasma etching.The 6 μm thick sacrificial photoresist (AZ 4562) was removed from below the cap structures by O2 plasma using a Matrix resist stripper (400 W power, 16 min etch time, 200 °C temperature).Figure 1 shows a SEM image of CCFF perforated membranes.Membranes had circular holes 5 μm in diameter, a non-perforated edge frame about 13-15 μm wide and porosity, given by the ratio between perforated area and total area of the membrane, ranging between about 8% and 32%, for various geometries.A kla-Tencor P-6 profilometer was used to measure the internal stress of each deposited layer.The wafer curvature was evaluated recording three profiles for each sample, then the internal stress of the tri-layer was calculated by Stoney's formula.The following values of stress were measured: −108 MPa (2 MPa std.deviation), compressive, for the silicon nitride layer and +261 MPa (1 MPa std.deviation), tensile, for the a-Si layer.The residual stress of the tri-layer was calculated equal to about +63 MPa, tensile.

Deflection Model of CCFF Membranes
In terms of maximum deflection amplitude, ωmax, under a given applied load, a perforated membrane deflects as much as an unperforated one of lower Young's modulus, where the proportionality's factor depends on the porosity.However, E is an intrinsic property of the material, invariant with respect to perforation.Conversely, the internal stress of the thin film can be likely affected by perforation, or in wider terms by micromachining [5].
The approximate deflection law proposed in this work defines the maximum deflection experienced by a CCFF membrane under increasing loads.Finite element parameter sweeping analysis allows to study the dependence of ωmax on the main mechanical and geometric parameters: length between the clamped-edges (L), width between the free-edges (W), thickness (h), Young's modulus (E) and residual stress (σm) of the membrane.For no-stress state, the maximum displacement is written as: For non-negligible stress, the following deflection law was derived: with: ω0, t and σm are free fit coefficients.The term ω0 is the deflection without loading, due to the state of internal stress of the membrane, represented by σm.

Results and Discussion
Deflection measurements under loads in the range 4.9-24.5 μN were performed with a Kla Tencor P-6 profilometer.Five CCFF membranes with different sizes and porosity were selected for the investigation.Figure 2a shows the deflection profiles measured on a perforated membrane of sizes W × L = 200 × 300 μm 2 and porosity 8.38%, under increasing loads.Since the perforation doesn't modify the elastic modulus of the material, the value measured by nanoindentation (136.6 GPa) was also considered meaningful for the perforated membranes and set as constant in the Equation (2).Therefore, deflection data were interpolated by the Equation (2) for calculating the internal stress of each membrane.Such a value was compared with that obtained by FEA, in which parametric sweep simulations were performed to identify the value of stress that returns the best agreement between predicted and measured maximum deflection data (Figure 2b-d).The intrinsic displacement ω0 was in the range 70 ÷ 100 nm, for all the investigated cases, as indicated by FEA simulations.The values of internal stress obtained by FEA and interpolation were comparable, in the range 10-18 MPa, with relative in the range 2.1-8.5%, for membranes at lower porosity.The reduction with respect to the value measured on the continuous tri-layer was due to the perforation and structuration of the tri-layer in form of clamped-clamped membrane.
Results demonstrated that the Equation ( 2) is efficient for providing an estimation of the internal stress of CCFF perforated membranes, starting from experimental data of load-deflection obtained by a stylus profiler.In complementary way, also the Young's modulus of CCFF perforated membranes can be extracted by the interpolation.

Figure 2 .
Figure 2. (a) Deflection profiles measured on a perforated membrane of sizes W × L = 200 × 300 μm 2 and porosity 8.38%, under increasing loads; (b) FEA results of displacement under quasi point load of 24.5 μN applied at the center of a membrane of sizes W × L = 200 × 300 μm 2 , porosity of 8.38%.Predicted versus measured maximum deflection of different perforated membranes: (c) membrane W × L = 200 × 300 μm 2 with increasing porosity; (d) three different membranes with same porosity.