Behavioral Modeling and Simulation of Micromechanical Resonator for Communications Applications

Cannes-Mandelieu, 5-7 May 2003 Behavioral Modeling and Simulation of Micromechanical Resonator for Communications Applications Cecile Mandelbaum, Sebastien Cases, David Bensaude, Laurent Basteres, and Philippe Nachtergaele MEMSCAP S.A., Parc des Fontaines, Bernin, 38926 Crolles Cedex, FR ABSTRACT A method for ing and simulating MEMS is presented for communications applications. This method includes the automatic generation of a simulation-ready description of the MEMS device including coupled electro-mechanical behavior extracted from a geometrical device description. In order to solve the lack of interoperability with simulators to perform MEMS analyses, the proposed method introduces a output fully compatible with behavioral simulators such as those from Tanner Research, Mentor Graphics, Synopsys, Agilent Technologies and Cadence for electronic simulations. This paper especially focuses on how to integrate a coupled-electromechanical in an RF simulator like ADS from Agilent. The complete design flow is presented including layout design, automatic 3D generation for 3D analysis, behavioral generation and integration of the component into a circuit simulation. Two methods are presented- automatic generation of an electrical equivalent circuit for the MEMS device and- a method creating a fully non-linear device which can be used as a black box within the circuit simulation environment. The paper presents an example based on a 10-MHz micromechanical resonator embedded within a Pierce oscillator circuit following the work of Nguyen [1]. The of the micromechanical resonator, a clamped-clamped beam that vibrates in a vertical displacement in response to an electrostatic excitation, is automatically created following the complete flow described above. The resonator has been analyzed at two levels: the device- and the systemlevel. On one hand, the intrinsic mechanical properties are obtained with the finite element method. On the other hand, the behavioral- and electrical equivalent circuits are generated from the finite element by reducing the number of degrees of freedom. The analysis of the oscillator circuit is then performed with several RF simulators. In Nguyen s work, the author has simulated the oscillator circuit using an RLC-equivalent of the micromechanical resonator. In contrast to the method presented in Nguyen article, the inclusion of coupled electro-mechanical behavior in a circuit level simulation allows the representation of the non-linear effects of the MEMS device. The simulation results, highly coherent with results obtained on the circuit that integrates the RLC-equivalent, highlight this mechanical non-linearity. 1. INTRODUCTION Due to MEMS interdisciplinary nature, MEMS design implies multi-physics knowledge and experts in various domains. It often leads to a lack of communications between physics engineering teams and electronic designers. The tools, often dedicated to one specific application, are not aimed at connecting different physics purposes. Therefore, the CAD output formats do not always match, especially in RF applications due to their specific range of frequency. This article deals with a ing method integrating the MEMS device in a system-level for RF simulation. First of all the main methodology will be described. Then, a test case regarding a 10MHz microresonator embedded in a Pierce oscillator [1] will be applied using CAD tools: MEMS Pro and MEMS Modeler from MEMSCAP for the ing, and ADS from Agilent and SpectreRF from Cadence for RF simulation. This application will demonstrate that reduced order s can be used for RF simulations and also highlight the remaining difficulties for a complete automatic integration. 2. CONCEPT OF MODELING AND ITS USE 2.1 Top-Down flow methodology for the MEMS design automation Figure 1 describes the MEMS design top-down flow that will be applied for the generation. The MEMS design starts with the validation of a concept that must fit all the required specifications. The aim is to size the device to get a prototype in a short time. These pre-dimensionings are often based on system-level simulations, dedicated to final applications such as

Concept Validation Mask Layout Design 3-Dimensional Model Analysis Mask Layout Completion Finishing System Specifications Systtem Environment - Multi-Physics Domain Model l Environment -- Multi-Physics Domain Manufacturing Fig. 1: Top-down design flow Desiign Automation Reduction Models Generator communications. The system-level circuit includes the MEMS device and its associated micro-electronic driving system. The next step is to design the mask layout from parameters and dimensions defined in the concept validation. The device may have to be modified according to the process constraints. A 3- dimensional involving coupled multi-physics is then generated from the mask layout to perform finite element analysis. This step may again modify the device design until final specifications are met. At last, a layout generated from the updated 3-dimensional is completed, ready to be sent to manufacturers. This method is efficient only if the step based on the 3D is shortened. Indeed, as MEMS s are complex structures dealing with coupled physics, their respective 3D s are also complex and finite element simulations are computationally expensive and time-consuming. As a consequence, the concept validation step is essential. 2.2 Models and system simulation of MEMS component The concept validation step requires powerful simulators such as Matlab or those coming from the IC domain such as Spectre from Cadence or T-SPICE from Tanner Research. The aim of this study is to create s fully compatible with these simulators. There are two standardized ways to and integrate an electro-structural MEMS component in a system-level simulation. The first one consists of using a sub-circuit with RLC components to describe the MEMS component. The second method consists of describing the behavior with an A-HDL language. Then, this can be included in a schematic for system-level simulation. The first method is easy to implement but it is then difficult to determine the values of the lumped components that will correctly describe the physical components. The second method is more adapted to describe the physical phenomena. The most direct technique is to write equations in the simulator language that represents the behavioral. This method assumes that the user has already a high knowledge of the language. It is also possible to use CAD tools for automatically generating a behavioral translated in specific IC languages such as A-HDL. This method is already used with standard IC simulators. But targeting an RF simulator leads to two problems: 1- the language is not always compatible; 2- the high frequency behavior of the device is completely different from the low frequency behavior. In this article the second method is applied and implemented into two different RF simulators. The will be described in an equivalent circuit (SPICE) to fit ADS format, and will also be described with behavioral equations (Verilog_A) to fit SpectreRF format. The results are compared with the corresponding RLC sub-circuit. 3. PRESENTATION OF THE TEST CASE This test case describing a micromechanical resonator embedded in a Pierce reference oscillator is similar to the one used from the work of Nguyen [1]. 3.1 The oscillator design This oscillator (Fig. 2) is roughly composed of 2 bipolar transistors in a Darlington stage and connected to a microelectromechanical resonator to set the oscillator frequency. This circuit can oscillate under the following condition: the sum of the overall loop phase shift of the circuit ( 180 o + f + f + f ) must be equal to 0 (or 360 ). 1 2 3 Focusing on the conceptual schematic of the oscillator (Fig. 3) this circuit can be compared to a steady-state transconductance amplifier connected to two RC-systems in parallel and connected to the micromechanical resonator. 3.2. Micromechanical resonator design The micromechanical resonator is a clamped-clamped beam over an electrode made in silicon surrounded by the air. Table 1 summarizes the beam dimension and the material properties.

Fig. 4: µresonator layout and its 3D view Z Y X Anchor Fig. 5: Model in ANSYS meshed with its boundary conditions Displacement of node N_Master node o 180 + f 1 + f 2 + f 3 = 360 Fig. 2: Modified Pierce oscillator Fig. 3: Conceptual schematic of the Pierce oscillator 4. BEHAVIORAL MODEL GENERATION The generation used follows the top-down method described previously. 4.1. Process and layout The first phase of this example is the layout design. The process is based on a customized PolyMUMPs process with 2-layer polysilicon micromachining. It is composed of a non-patternable nitride isolation layer, a polysilicon ground (plane) layer, two structural polysilicon layers and one oxide release layer. The 3D (Fig. 4) is automatically generated from a layout in the MEMS Pro design suite from MEMSCAP. 4.2. 3D o Beam dimension 40* 8* 2 (µm): L, W, t Electrode: L, W 20*10 µm 2 Air Gap 0.1 µm Si Modulus 160 GPa Young Si Poisson s 0.26 ratio Si Density 2330 Kg/m 3 Air Relative 1 Permittivity Table 1: Properties of the microresonator In MEMS Pro, a part of the mask layout (the beam without the anchors and the oxide above the electrodes) is translated into a.sat file for ANSYS compatibility. Material properties, meshing and boundary conditions are added (Fig. 5) to the 3D. Meshing elements are tetrahedrons compatible with structural and electrostatic simulations. The anchors are ed by setting the appropriate mechanical boundary conditions in ANSYS. 4.3. Model reduction The Model Deformation Algorithm from MEMS Modeler, MEMSCAP software, is used to generate behavioral s for an electrostatic structural coupled systems reduction. It allows the reduction of a finite element into a few degrees of freedom related to nodes named N_Master. The finite element includes mechanical and electrostatic properties. Therefore, the reduction can take into account coupling effects. Here, it is reduced to one degree of freedom, which is related to the N_Master node situated in the middle of the beam. The reduced describes the behavior of this node in a limited range of displacement: the beam is moving up and down (Fig. 5). The reduced output is described by polynomial equations relative to the dynamics: 2 d x dx M + D + Kx = F (1) 2 cpl M: reduced mass matrix D: reduced damping matrix K: reduced stiffness matrix F cpl : coupled force (external), here due to the electrostatic force. X: displacement on the N_Master node These equations are written in two languages supported by the two selected RF simulators and the associated output s are also different. Written in SPICE, one is described into an equivalent circuit with capacitance, inductance, resistance, controlled source The input is a differential voltage. The output is the capacitance, the speed, the acceleration and the displacement of the microresonator. The second equations are written in Verilog-A. This contains the behavior of the system with 3 pin outputs: two electric

C 0 15 ff R 120 ko L 0.101H C 2.488fF Fig. 6: The referenced RLC Fig. 7: SPICE netlist in ADS pins describe the electrostatic behavior of the two conductors and one mechanical pin describes the mechanical behavior (the displacement and the force feedback). Each pin represents a dual value. The across value of the electrical pin corresponds to the voltage and the through value to the current that flows through the. The across value of the mechanical pin corresponds to the displacement of the degree of freedom and the through value to the force. These 2 previous s are compared to a RLC described in the Figure 6. 5. INTEGRATION INTO SYSTEM LEVEL As ADS from Agilent can import a written in SPICE, the reduced by the MEMS Modeler tool is used for this simulator. The written in Verilog-A is used with Spectre from Cadence. 5.1. Importing the SPICE into ADS 5.1.1. Step 1: Importing the SPICE into ADS The ADS netlist translator performs an automatic translation of the SPICE file into an ADS netlist (Fig. 7). In this operation, the number of pins must be specified. Two pins are needed, one relative to the input voltage (input 1) and the other to the capacitance value (output 1). 5.1.2. Step 2: Modifying the ADS netlist The netlist syntax imported in ADS must be slightly modified (the addition of some quotes ). Controlled source devices are ill managed by ADS and must be replaced by SDD devices (Symbolically-Defined Devices) [2]. SDD devices are user-defined nonlinear components that can allow non-linear devices simulation in both largesignal and small-signal analyses. The second advantage is that implicit and explicit equations can be defined. Then equations relate variable ports (current port and voltage port) and their derivatives. As an example: F(1,0)=a* _V2 (a is a constant and _V2 the voltage on port V2). Fig. 8: SDD device used in ADS for SPICE importation F(1,1)= _V3 b (b is a constant, and _V3 the voltage on port V3) It means that F(1,1)= 1 d_v3. b So, the implicit function F (1) on port 1 is equal to: F(1)=F(1,0)+ F(1,1)= a*_v2+ 1 d_v3 =0 (2) b 5.1.3. Step 3: Including the ADS netlist in the circuit The output of the cannot be directly connected to the electrical circuit. The ADS schematic must be modified to add the capacitor behavior that links the voltage to the current: d(cv) d(v) d(c) I = = C + V (3) The ADS is then associated with an SDD device that allows the calculation of the current versus the input voltage and the connection to the oscillator circuit. This SDD contains 4 ports (Fig. 9). Port 1 has 2 outputs in and out that will be connected to the main oscillator circuit. Port in will be connected to input1 port of the ADS netlist. As a consequence, the voltage V1 on port 1 will be the input voltage needed for the microresonator. Port 3 will be connected to output1 of the ADS netlist that contains the capacitance value of the microresonator. So, the voltage on this port is equal to the capacity value of the microresonator. _V 3 = Cµ resonator (4) Ports 2 and 4 are defined through implicit equations: On port 2, the voltage _V2 is a function of the derivative of the input voltage _V1: On port 4: F(2) = F(2,0) + F(2,1) = 0 d_v1 V + = 0 => _ V2 = (5) d_v 1 _ 2 F(4) = F(4,0) + F(4,1) = 0 (7) d_v d_c 3 µ _ V4 = => _ V4 = resonator (6) (8)

f 0 ANSYS simulator 10.7MH z Spectre simulator Behavior RLC al 11 MHz 10.1 MHz ADS Simulator Behavioral RLC 10.9MHz 10.1 MHz Table 2: First Frequency of the microresonator Harmonic Behavioral RLC f 1 11.1MHz 10.6MHz f 2 22.1MHz 22.2MHz f 3 33.1MHz 31.8MHz f 4 44.1MHz 42.4MHz f 5 55.2MHz 53.1MHz Harmonic Behaviora l RLC f 1 10.5MHz 10.1MHz f 2 20.9MHz 20.2MHz f 3 31.5MHz 30.2MHz f 4 41.9MHz 40.3MHz f 5 52.3MHz 50.3MHz Table 3: First harmonic frequencies of the oscillator after harmonic balance simulation performed with ADS According to equations (4), (6) and (8), the current on port 1 is: I(1) = I(1,0) + I(1,0) (9) I(1) = Cap* _V2 + 1e 12* _V 1 * _V4 (10) d_v1 d_v3 I(1) = 1e 12(_V 3 * + _V1 * ) (11) dvµ resonator dc µ resonator I(1) = 1e 12(Cµ resonator * + V * ) µ resonator (12) The current and the voltage are related to a capacitor behavior. The component is ready to be included in the oscillator circuit between ports in and out. 5.2. Importing the Verilog-A into Spectre The Verilog-A is automatically imported in Spectre without any change. A symbol is associated with this netlist. The component has three pins as described in section 4.3. The two electrical pins are connected to the circuit. The mechanical pin is connected to current source to a force. As a consequence, the microresonator freely oscillates. 6. RESULTS 6.1. Modal simulation of the standalone resonator Table 2 compares the first frequency of the standalone microresonator s (RLC and behavioral s) using different simulators (ANSYS, Spectre and ADS). The finite element frequency is slightly higher than the RLC equivalent one. The first frequency of behavioral is similar to the finite element one. Table 4: First harmonic frequencies of the oscillator after harmonic balance simulation performed with SpectreRF 6.2. Harmonic balance simulation of the Pierce oscillator Harmonic balance simulations can handle non-linear devices such as the microresonator. Table 3 and Table 4 compare first harmonics of the Pierce oscillator including different s (RLC and behavioral s) extracted from harmonic balanced simulation. All s have an oscillation frequency around 10-11MHz. The circuit simulated with SpectreRF with the referenced RLC presents a higher resonance (10.6MHz) than expected. This is due to the used bipolar transistor s that are defined differently in the Agilent oscillator circuit and the Cadence one. Therefore, it is difficult to obtain similar circuits. In ADS, the behavioral seems to affect the overall system and shows a lower frequency than expected. Figure 9 represents the phase noise on the input and output of the system simulated with ADS with an offset from the carrier. The phase noise of the system is defined by two curves: the (phase noise for mixing analyses) and the (phase noise for frequency sensitivity analyses). The is more sensitive and therefore reliable near the carrier frequency and the is more reliable far away from the carrier frequency. If the phase noise including the reference RLC is highly linear, the plots of the phase noise including the written in SPICE present additional peaks at 20kHz from the carrier characterizing non-linearity phenomena. This is reinforced as the curve starts above the curve, cuts it and finishes below it. Figure 10 is the plot of the phase noise with an offset from the carrier obtained with Cadence. The same kind of results is obtained. Simulation results on a circuit including a referenced RLC are linear, and simulation results including behavioral presents additional peaks around 20kHz from the carrier. This means that these effects are coming from the behavioral s. It is related to the non-linear mechanical properties of the microresonator. In order to highlight the slopes of the phase noise, Figure 11 shows the phase noise of the system including the behavioral in the

Phase noise At the Input Phase noise At the RLC Model Behavioral RLC Model Behavioral Model Non-linear Peaks Input Phase Output Phase Noise Input Phase Non-linear Peaks Output Phase Noise Fig.10: Phase noise of the oscillator at the input and at the output of the circuit simulated with SpectreRF Fig.9: Phase noise of the oscillator at the input and at the output of the circuit simulated with ADS logarithmic scale. The phase noise curves have 2 main slopes. In a first part, it starts with a slope of 1/f 2, when the non-linear peaks appear the slope is modified into 1/f 3. The curves have not yet reached stability at 100kHz that is unexpected and not coherent with the referenced article of N Guyen. Additional works should be done to better understand the phase noise behavior of mechanical s. Fig.11: Phase noise of the behavioral with a logarithmic scale simulated with SpectreRF Non-linear Peaks Output Phase Noise Input Phase Noise 1/f 3 7. CONCLUSION Reduced s from microresonator finite element s have been translated into circuit-equivalent and behavioral s and then integrated in a system-level simulation for RF. The Verilog-A has almost been automatically imported. The SPICE still requires a lot of modifications for a correct insertion in the main circuit. On the whole, the results are satisfying but this work emphasizes some discrepancies between simulators (s definition). Comparing to the referenced RLC, it has been shown that including a behavioral highlights the non-linearity due to the physical properties of the micro-systems. The next step will be to compare these results to experimental data. Additional study would be to focus on behavioral and reduce the finite element into several degrees of freedom relevant to a 10MHz behavior. 11. REFERENCES [1] Clark T.-C. Nguyen, et al., A 10-MHz micromechanical resonator Pierce reference oscillator for communications, digest of Technical Paper, Transducers 01, Germany, pp. 1094-1097, 2001. [2] Agilent Technologies, Advanced Design System 2002- Analog/RF User-Defined Models, Agilent Technologies, February 2002 [3] MEMSCAP, MEMS Pro User Guide, 2002 [4] Clark T.-C. Nguyen, Frequency-selective MEMS for Miniaturized Communications Device, 1998 IEEE Aerospace conference, Snowmass, Colorado, vol. 1, pp. 445-460, 1998 [5] Clark T.-C. Nguyen, Micromechanical Circuits for Communication Transceivers, 2000 Bipolar/CMOS Circuits and Technology Meetings, Minneapolis, Minnesota, pp. 142-149, 2000 [6] Clark T.-C. Nguyen, High-Q Micromechanical Resonators and Filters for Communications, 1997 IEEE int. Symposium on Circuits and Systems, Hong Kong, vol. 1, pp. 2825-2828, 1997 [7] Clark T.-C. Nguyen, Micromechanical for Resonators and Filters, 1995 IEEE int. Ultrasonic Symposium, Seattle, pp. 489-499, 1995 [8] Reza Navid, et al., Third-order Intermodulation Distortion in capacitively-driven cc-beam micromechanical for resonators, Technical Digest, Int. IEEE Conf. Micro Mechanical Systems Conference, Interlaken, Switzerland, pp. 228-231, 2001