Numerical Simulation and Optimization of Capacitive ... - inuTech

Numerical Simulation and Optimization of Capacitive Transducers F. Vogel§, H. Landes#, R. Lerch*, M. Kaltenbacher*, R. Peipp* §

- inuTech GmbH, Fürther Str. 212, 90429 Nuremberg, Germany # - WisSoft, Eisenstr. 35, 91054 Buckenhof, Germany * - Department of Sensor Technology, University of Erlangen-Nuremberg, Paul-Gordan-Str. 3-5, 91052 Erlangen, Germany

Abstract A recently developed calculation scheme which supports the automated computer based design of capacitive electromechanic transducers is presented. The simulation environment consists of two modules: one for the precise simulation of the coupled electro-mechanicacoustic system and the second providing flexible and powerful optimization algorithms. The capabilities of this new simulation scheme are demonstrated by considering several industrial application examples: cMUTs, acceleration sensors, and a CMOS microphone. 1. Introduction Capacitive transducers have recently found an increasing interest in industrial applications. Due to small size and the possibility of integrating signal processing electronics on chip, these devices promise to be an attractive alternative to transducers fabricated with classical technologies. As a consequence, an increasing demand for computer tools to precisely simulate the transducer behavior is observed in order to speed up the design process and the development cycle of such devices. To meet these needs, we have recently improved significantly the available algorithms in our simulation package CAPA [1]. Based on combinations of Finite and Boundary Element Methods (FEM/BEM) the motion and deformation of an electrically charged elastic body in an electric field is calculated. Furthermore, the influence of the deformation and translation of the body on the electric field is also taken into account. However, even with the precise simulation of the transducer behavior available, the design and optimization process of a specific device still involves a huge amount of user interference. Therefore, we have recently implemented a coupling of CAPA to powerful numerical optimization modules. Several approaches for material as well as geometry optimization are available. Therewith, an automated optimization subject to process or externally specified restrictions can be established, minimizing user interference and significantly speeding up the overall development process. The rest of the paper is organized as follows. First we will shortly recall the underlying principles and the available formulations of the simulation software. Both FEM and combined FEM/BEM approaches will be presented. Next, the necessary background of the optimization software as well as the available features are discussed. Finally, several applications are used to demonstrate the applicability of our approach: an experimental micromachined capacitive ultrasound

transducer array (cMUT), the design of comb structures for use in acceleration sensors, and the optimization of an electrostatic membrane device for an integrated silicone microphone. 2. FEM-BEM Theory The precise simulation of capacitive transducers requires the coupled solution of the mechanical and electrostatic fields, and, in many cases, the coupling to an ambient fluid medium also has to be taken into account. In the case of an electrostatic-mechanical transducer the coupling between the electric and the mechanical field is caused by the electrostatic force between the electrodes. Denoting by E = (Ex , Ey , Ez) the electric field, this force is calculated by means of the electrostatic force tensor, which is given by

The electrostatic force FE is given by

where n denotes the normal vector on the surface A. This force leads to a deformation of the electrodes, which has to be considered in the electrostatic simulation. Splitting the coupled system into mechanical-acoustic and electric parts and applying the FE method analogous to [2] we obtain the matrix equations

In equations (2,3), u, φ, and ψ denote mechanical displacement, electric, and acoustic velocity potential, respectively. Muu, Cuu, and Kuu stand for the mechanical mass , damping, and stiffness matrices, whereas Mψψ, Kψψ, and Cuψ are acoustic mass, stiffness and acousticstructure coupling matrices. Kφ denotes the electric stiffness matrix, Fu(φ) the nodal vector of external mechanical and electrostatic forces, and Fφ(u) the electric source vector. It should be noted, that both load vectors depend on the current state of the mechanical displacements and the electric potentials, respectively.

The time discretization is performed by applying the Newmark method to the mechanical-acoustic system (2) and a moving mesh technique is used to avoid large mesh deformations in the FE grid for the electrostatic field computation [3]. As an alternative to the pure finite element approach a coupled FEM-BEM simulation scheme to coupled electrostatic-mechanical transducers is also available. In this case the electrostatic field is treated by boundary elements resulting in the matrix equation with the two boundary element matrices Hφ and Gφ, the nodal vector Φ of the scalar electric potential and the nodal vector En of the normal component of the electric field. This approach has the advantage that deformations of the electrodes will not cause a deformation of surrounding finite elements which are necessary to describe the electric field in the case of pure FE modeling. As the boundary element matrices are updated corresponding to the mechanical displacement, the direct coupling of (4) and (2) leads to a nonlinear system of equations. Using predictor values in combination with a predictormulticorrector algorithm a decoupling of the matrix equations can be achieved [4]. In order to efficiently evaluate the boundary element matrices, which is strictly required for the coupled system simulation, a multipole approach [5] has been implemented recently. 3. Optimization Most real-life design optimization problems require the simultaneous optimization of several possibly conflicting objective functions. These multi-objective optimization problems [6] can be mathematically formulated as follows: where k≥2 and

denotes the set of feasible solutions, which is assumed to be non-empty. In contrast to single-objective (scalar) optimization problems, in multi-objective optimization problems there is usually no solution x* which is optimal for all objective functions simultaneously. Thus, relaxed optimality conditions are required. One approach to overcome this difficulty relies on the notion of pareto optimality. Here, a point x* is called pareto optimal for (MOP), if no criterion can be improved without worsening at least one other criterion. Usually, this notion of optimality leads to multiple optimal solutions, the so called pareto front. The multi-objective problem is almost always transferred into one or a sequence of single objective or scalar optimization problems (scalarization) whose

solutions are pareto optimal for problem (MOP). Those scalar problems are then solved by efficient and reliable algorithms for single objective nonlinear optimization. In the optimization modules used in our software environment, numerical methods for multi-objective optimization based on a priori and a posteriori preferences are available (for a complete characterization of available preference methods see [7]). A posteriori methods are methods for generating pareto optimal solutions by scalarization after the pareto set (or a part of it) has been generated. One option is the so called ε-constraint method, where one of the objective functions is selected to be optimized and the other objective functions are converted into constraints by setting an upper bound to each of them. A priori methods are methods where the preference must be specified before the solution process. One possibility is the so called preference or value function approach, where an accurate and explicit mathematical formulation of a scalar preference or value function which represents the preference globally must be given. The difficulty is that for most problems the engineer does not necessarily know beforehand what is possible to attain in the problem and how realistic her or his expectations are. As indicated above most deterministic mathematical schemes for the solution of multi-objective optimization problems are based on solving one or a sequence of single objective optimization problems. Several different categories of optimization algorithms can be distinguished [8]. Due to its efficiency and robustness (especially for the problems considered in the context here), we will only look at so-called sequential quadratic programming (SQP) methods. These have been established as the standard general purpose tool for solving smooth nonlinear optimization problems under the following assumptions: • the problem is not too large • the functions and gradients can be evaluated with sufficiently high precision • the problem is smooth and well-scaled • there is no further model structure that can be exploited. Detailed discussions about SQP methods are contained in [9], and the numerical comparative studies contained in [10] show their superiority over other mathematical programming algorithms under the assumptions mentioned above. The fundamental idea behind SQP is to generate a sequence of quadratic programming subproblems obtained by a quadratic approximation of the Lagrangian function and a linearization of the constraints. Second order information is updated by a quasi-Newton formula and the method is stabilized by an additional line search. Among the most attractive features of SQP methods is the superlinear convergence speed in the neighborhood of a solution.

As an application of our simulation scheme we consider an ultrasound transducer array as shown in fig. 1. During measurements the above described drawbacks like crosstalk between neighboring elements as well as long ring down times have been observed. A finite element model of single transducer array, consisting of 19 membranes was established (fig. 2). Due to symmetry considerations, only a quarter of the array had to be modeled. The membranes had a thickness of 1 µm and the gap between the electrodes was 500 nm. In the simulations the use of an integrated controller to improve ring-down behavior and cross-talk was investigated.

Fig. 1: Setup of an experimental cMUT array 5. Micromachined Ultrasound Transducers Capacitive micromachined ultrasound transducers have recently found increasing attraction in medical applications. Due to their ease of fabrication and integration with electronics, they promise to be an attractive alternative to the standard piezoelectric transducers. Especially in the design and development of ultrasound phased array antennas the use of cMUTs is currently strongly investigated, as several shortcomings of this technology have still to be overcome (strong crosstalk, long ring-down times). Due to the large deformations, which the membranes may observe, not only the nonlinearity due to electrostatic coupling force, but also geometric nonlinearities in the mechanical part have to be considered. In principle, this results in an additional nonlinear iteration loop inside the mechanical solution part, therewith further increasing solution time. To overcome this shortcoming, we have recently developed a so called working-point method [11], in which a linearization around the working point of the membrane is performed. Therewith, significant speedup by a factor of 100 and more is obtained and the simulation of a full three-dimensional setup becomes feasible.

Fig. 2: Finite element model for cMUT array

Fig. 3: Center displacement of controlled and uncontrolled membrane The controller applies a control voltage, which is calculated based on the current center displacement of the membrane, to the electrode. In the case of a single period sine burst electric excitation (10 V amplitude), the ring down time can be significantly improved. This is shown in fig. 3 for the case that all membranes are excited in parallel. As can be seen from fig. 4, this improvement also shows up in the radiated sound pressure signal, whereas in fig. 5 the smoothing effect of the controller on the pressure pulse spectrum is shown.

Fig. 4: Pressure pulse signal for uncontrolled and controlled cMUT array

leads to significantly reduced mesh generation times, especially in the case when the position of the electrodes results in a very small gap (near contact). A detail of a coarse FEM/BEM model is shown in fig. 7, where the

Fig. 5: Smoothing effect of controller on pressure pulse spectrum 6. Acceleration sensor In this application we consider an acceleration sensor, where so-called comb structures play an important role. The principle setup of the sensor element is shown in fig. 6. A plate, which is usually made of silicon, is mounted on 4 flexible spring elements. Comb structures, consisting of rows of electroded teeth, are formed on the boundary of the plate, interlocking with stationary counterparts along the sides. Due to the flexible mounting of the plate, the relative position of the movable and stationary electrodes will change in case of an acceleration. The resulting capacity deviation from the initial one for the structure at rest can be detected and used as a measure for the deformation and therewith for the acceleration. Due to the small size of the moveable and deformable parts (low inertial mass) the sensor shows a very fast response time.

Fig. 7: Interlocking teeth of acceleration sensor (detail of coarse FEM/BEM model) interlocking of the stationary and the moveable electrodes is clearly visible. Each tooth had a length of 218 µm and a thickness of 3 µm, whereas the initial gap between 2 interlocking teeth was set to 4.5 µm. The thickness of the plate was 50 µm. In fig. 8 the calculated capacities of the comb structure for several mesh sizes are plotted as a function of an applied horizontal offset, i.e. the moveable electrodes were displaced within plane with the stationary ones. It can be observed that FEM and FEM/BEM simulations produce nearly equivalent results and also the strong increase in the capacity near the contact condition for an offset of 4.5 µm is reflected very nicely. In fig. 9 the relative deviation of the simulation results from the theoretical curve, in which stray field effects are neglected, have been depicted. Here, the influence of the mesh size becomes more obvious.

Fig. 6: Principle setup of acceleration sensor In a sequence of simulation runs, the change in the capacity due to position changes was calculated both for the pure FEM as well as for the coupled FEM/BEM approach. For the first approach every calculation requires the full meshing of the resulting gap between the electrodes. In contrast to this, for the coupled FEM/BEM approach the meshing of the gap is no longer required, since the electric field between the electrodes is modeled by means of boundary elements. Therewith, this approach

Fig. 8: Capacity for comb electrodes as a function of inplane offset

seriously. In order to increase the sensitivity, several design modifications besides the simple circular plate have been proposed, like finger or corrugated membranes. Here, we will only consider finger type membranes and the principle setup of such a membrane is shown in fig. 11. Each membrane is defined by its circumference radius R, the inner radius R0 of the membrane, the membrane thickness t, the number of fingers n as well as the width b of the fingers. Only membranes with 4, 8, and 16 fingers have been considered in the simulation runs.

Fig. 9: Relative deviation of simulation results from theoretical capacity (stray field neglected) 7. Optimization of a CMOS Microphone Membrane Integrated CMOS microphones [11] offer several advantages when compared with standard microphone setups. Miniaturization, high temperature stability, low power requirements, as well as low cost and mass production are only some issues, which are worth mentioning. The applicability of such devices, on the other hand, strongly depends on the sensitivity and the signal to noise ratio which can be achieved. The sensitivity of the microphone, in turn, depends on the corresponding one of the used membrane. In our final study presented here, we have therefore applied our newly developed combination of simulation and optimization software to the optimization of a special membrane design.

Fig. 10: Principle setup of a CMOS microphone The principle setup of the types of microphones we consider is shown in fig. 10. A flexible membrane electrode, which usually is made of polysilicon, is mounted along its boundary. A small air-filled gap separates the flexible electrode from the stationary counter electrode. Typical dimensions of such a device are membrane radius 500-1000 µm and a gap height of 100-200 nm. The flexible membrane is subject to considerable prestressing, and, therewith, the sensitivity of the membrane to pressure loads must be considered

Fig. 11: Principle setup of finger type membrane As the microphones are usually also required to have a large bandwidth, the design goal chosen consisted in the simultaneous optimization of bandwidth and sensitivity of the membrane. Therewith, this problem corresponds to a typical multi-objective optimization problem as discussed in section 4. In order to apply the methods described above, we had to choose an appropriate scalarization of this multi-objective optimization problem. For this it should be noted that the bandwidth is limited by the first resonance of the membrane, whereas the sensitivity can be described by the static deflection of the membrane due to a specified pressure. Therefore, we considered the negative product of the first eigenfrequency f1 of the membrane and the center deflection uc due to a fixed pressure as our scalar optimization function. This approach accounts to applying an a priori method as defined in section 4. For every membrane the circumference radius R as well as the thickness t have been fixed parameters during the optimization runs. The radius R was chosen to be 400 µm, 500 µm, 600 µm, 800 µm, and 1000 µm and, since we considered 4, 8, and 16 fingers, we ended up with a grand total of 15 scalar optimization problems:

These optimization problems have been solved employing a SQP method as part of the CAPA optimization module. Due to its fast local super-linear convergence behavior those problems have all been solved in only a few iterations. The sensitivities of the objective function with respect to the optimization parameters, as needed by SQP during the optimization process, are calculated semi-analytically inside the CAPA software. In a large number of simulation runs, with nearly no user interference required, the solutions of all optimization problems have been calculated.

In table 1 the upper and lower bounds bl, bu, R0l, and R0 on the optimization variables are stated for the 5 optimization problems for the 4 finger membrane, whereas in table 2 the obtained optimal solutions with corresponding values for the first eigenfrequency (f1opt) and center displacement (ucopt) are given. In fig. 12 the maximum product values are depicted as a function of the radius R for all number of fingers considered. u

R 400 µm

bl 10 µm

bu 20 µm

500 µm

10 µm

20 µm

600 µm

10 µm

20 µm

800 µm

10 µm

20 µm

1000 µm

10 µm

20 µm

R0l 170 µm 200 µm 230 µm 300 µm 380 µm

R0u 320µm 350 µm 420 µm 560 µm 700 µm

Table 1: lower and upper bound for optimization parameters, 4 finger membrane R 400 µm 500 µm 600 µm 800 µm 1000 µm

R0opt 248 µm 276 µm 299 µm 304 µm 384 µm

bopt 10 µm 10 µm 10 µm 10 µm 10 µm

f1opt 44.9 kHz 36.3 kHz 22.6 kHz 10.3 kHz 6.42 kHz

Fig. 13: Pareto curve for 4 finger membrane with R=500 µm (frequency given in kHz).

ucopt 0.8 µm 1.3 µm 2.9 µm 13 µm 34 µm

Table 2: Optimum values for 4 finger membrane

Finally it should be noted, that in the above given optimization results stress-stiffening effects have not been taken into account. These effects, of course, must be considered for a realistic membrane optimization, as they play a significant role especially for large deformations. 8. Conclusions We have presented applications of our software environment to the numerical simulation and optimization of electrostatic transducers. This environment consists of the FEM/BEM software CAPA for the precise calculation of coupled electro-mechanic-acoustic systems and a recently implemented coupling to modules for the solution of complex optimization problems. Therewith, an automatic computer based design of all kind of transducer devices minimizing user interaction is made available. Acknowledgments The authors thank Marc Füldner and Dr. Robert Aigner from Infineon Technologies for fruitful discussions and helpful remarks on CMOS microphone technology.

Fig. 12: Optimum values of the scalar optimization function for 4, 8, and 16 finger membranes (R in mm) As an example of a pareto front, the ε-constraint method described in section 4 above is applied for a 4 finger membrane with R=500 µm. The resulting pareto curve is shown in fig. 13 and gives the static center deflection as a function of the first eigenfrequency, or, in other terms, the sensitivity of the membrane as a function of the bandwidth.

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