Fast Simulation of Electro-Thermal MEMS: Efficient Dynamic Compact Models (Microtechnology and MEMS)

Initial conditions and boundary conditions are to be set up within the interface. Next, you define the mesh and select a solver. Finally, you visualize the results and process and export the results. The solvers are set up automatically with default settings, which are already tuned for each specific interface. However, the advanced user can access and modify low-level solver settings as needed. Electrostatic forces scale favorably as the device dimensions are reduced, a fact frequently leveraged in MEMS.

The MEMS Module provides a dedicated physics interface for electromechanics that, for MEMS resonators, is used to compute the variation of the resonant frequency with applied DC bias — the frequency decreases with applied potential, due to the softening of the coupled electromechanical system. The small size of the device results in a MHz resonant frequency even for a simple flexural mode. In addition, the favorable scaling of the electromagnetic forces enables efficient capacitive actuation that would not be possible on the macroscale.

Furthermore, you have the option to use the electromechanics interface to include the effects of isotropic electrostriction. Piezoelectric forces also scale well as the device dimension is reduced. Furthermore, piezoelectric sensors and actuators are predominantly linear and do not consume DC power in operation.

Quartz frequency references can be considered the highest volume MEMS component currently in production — over 1 billion devices are manufactured per year. The physics interfaces of the MEMS Module are uniquely suitable for simulating quartz oscillators as well as a range of other piezoelectric devices. One of the tutorials shipped with the MEMS Module shows the mechanical response of a thickness shear quartz oscillator together with a series capacitance and its effect on the frequency response.

Thermal forces scale favorably in comparison to inertial forces.

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That makes microscopic thermal actuators fast enough to be useful on the microscale, although thermal actuators are typically slower than capacitive or piezoelectric actuators. Thermal actuators are also easy to integrate with semiconductor processes, although they usually consume large amounts of power compared to their electrostatic and piezoelectric counterparts. The MEMS Module can be used for Joule heating with thermal stress simulations that include details of the distribution of resistive losses. Thermal effects also play an important role in the manufacture of many commercial MEMS technologies with thermal stresses in deposited thin films that are critical for many applications.

The MEMS Module includes dedicated physics interfaces for thermal stress computations with extensive postprocessing and visualization capabilities, including stress and strain fields, principal stress and strain, equivalent stress, displacement fields, and more.

There is also tremendous flexibility to add user-defined equations and expressions to the system. For example, to model Joule heating in a structure with temperature-dependent elastic properties, simply enter in the elastic constants as a function of temperature — no scripting or coding is required. When COMSOL compiles the equations, the complex couplings generated by these user-defined expressions are automatically included in the equation system. The equations are then solved using the finite element method and a range of industrial strength solvers. Once a solution is obtained, a vast range of postprocessing tools are available to interrogate the data, and predefined plots are automatically generated to show the device response.

COMSOL offers the flexibility to evaluate a wide range of physical quantities, including predefined quantities like temperature, electric field, or stress tensor available through easy-to-use menus , as well as arbitrary user-defined expressions.

The Fluid-Structure Interaction FSI multiphysics interface combines fluid flow with solid mechanics to capture the interaction between the fluid and the solid structure. Solid Mechanics and Laminar Flow user interfaces model the solid and the fluid, respectively. The FSI couplings appear on the boundaries between the fluid and the solid, and can include both fluid pressure and viscous forces, as well as momentum transfer from the solid to the fluid — bidirectional FSI.

The MEMS module has specialized thin film damping physics interfaces which solve the Reynolds equation to determine the fluid velocity and pressure and the forces on the adjacent surfaces. These interfaces can be used to model squeeze film and slide film damping across a wide range of pressures rarefaction effects can be included. Thin-film damping is available on arbitrary surfaces in 3D and can be directly coupled to 3D solids. The ease of integration of small piezoresistors with standard semiconductor processes, along with the reasonably linear response of the sensor, has made this technology particularly important in the pressure sensor industry.

For modeling piezoresistive sensors, the MEMS Module provides several dedicated physics interfaces for piezoresistivity in solids or shells. The Solid Mechanics physics interface is used for stress analysis as well as general linear and nonlinear solid mechanics, solving for the displacements.

The MEMS Module includes linear elastic and linear viscoelastic material models, but you can supplement it with the Nonlinear Structural Materials Module to also include nonlinear material models. You can extend the material models with thermal expansion, damping, and initial stress and strain features. In addition, several sources of initial strains are allowed, making it possible to include arbitrary inelastic strain contributions stemming from multiple physical sources. We solve 1 with the initial condition T0 0 , the Dirichlet boundary condition T 0 at the bottom of computational domain.

Assuming that the heat generation is uniformly distributed within the heater and that the system matrices can be considered as temperature independent at the operation temperature, the finite element method FEM based spatial discretization of 1 leads to a large linear ordinary differential equations ODE system with nonlinear, i. Equation 3 is a starting point for model order reduction MOR , which leads to a system of the same form but with much smaller dimension, as schematically shown in Fig. Place of model order reduction within a conversion process from physical to compact model. As already mentioned in section 1.

There are two main algorithms belonging to Krylov-subspace-based moment matching approach, Arnoldi and Lanczos. Arnoldi algorithm matches r moments and Lanczos algorithm 2r moments, as described in more details in chapter 1.

Fast Simulation of Electro-Thermal MEMS : Efficient Dynamic Compact Models

The moment matching property is achieved by constructing an orthonormal projection matrix V R n r the construction process is described in chapter 1. As expected, frequency domain is best approximated around the expansion point s0 0. Comparison between the full-scale and the reduced order models of optical filter [2] in a single defined output node for the constant heating power of 1mW. The results in Fig. As there is a single heat-source voltage applied to the heating resistor , B in 3 is a column-vector.

Algorithm 3 from section 1. In more general cases, the temperature response in several or even in all finite element nodes might be required. In [1] it was demonstrated that, with the Block-Arnoldi algorithm from [5], it is possible to approximate not only a single output response but also the transient thermal response in all finite element nodes of the device. This is due to the fact that Block-Arnoldi does not take into account the output vector matrix C when constructing the basis V of the single projection subspace see section 1.

Mean square relative difference defined in [1] for all of the 1, finite element nodes of the optical filter [2] during the initial 0. It is further possible to transfer the nonlinearities 4 of the input function into the reduced system. Step response outer plot and step response error inner plot of the gas sensor [1] in a single output node for temperature dependent heating power according to 4 with 1. In order to apply the Arnoldi-based model order reduction, MEMS designer has to provide a discretized model e. This is done by choosing one or more expansion points in the frequency domain, as explained in section 1.

In [1], three heuristic methods to estimate the error of Block- Arnoldi-based reduction were suggested. Error indicator from [1] for the gas sensor model [1] with DOF. As MEMS are often composed of interconnected subsystems, array structures for example, it is desirable to reduce each subsystem on its own and then to couple them back together, following e. The main problem thereby is that the thermal flow is not lumped by nature as, for example, the electrical flow is along metallic wire interconnects.

The ratio of electrical conductivity of metals and that of insulators is of the order of Hence, the electrical current flow takes place almost solely in metal paths. This is not the case with heat flow because the ratio of thermal conductivities in microtechnology is only of the order of Therefore, it is unclear how to lump the thermal fluxes at shared surfaces between two finite element models in order to form the thermal ports which would serve to couple together several compact models.

Note that, if one would keep all the surface nodes as ports, i. Step response of the full-scale and reduced order models for the microhotplate array based upon a gas sensor device from [4]. Reduction is done by block Arnoldi from [5], in case when two heat sources of 40mW each are switched on. General technique of coupling two reduced thermal models gained by Krylov-subspace-based projection is discussed in [1].

In this respect, the structure-preserving model reduction techniques [6] should be further investigated. A silicon-nitride membrane with integrated heater and sensing element was fabricated by low-frequency plasma enhanced chemical vapor deposition. The microstructured hotplate consists of a thin film membrane composed of silicon nitride suspended over a silicon frame.

Thin film metal resistors are fabricated on top of the membrane for heating and temperature sensing. In order to achieve a preferably circular symmetric and homogenous temperature distribution at the center of the square membrane, both resistors are arranged as shown in Fig. Schematic view of the thin-film resistors for heating and temperature sensing. Heating resistor is operated at constant voltage. The sensing resistor is configured for four-point measurement.

The characterization of the static and transient thermal properties of the membrane is performed on a temperature controlled mount. The electrical resistance depends linearly on the temperature over the investigated temperature range and is modeled by 4.

The transient thermal response of the membrane is characterized by applying a rectangular voltage signal to the heating resistor using a function generator. The thermal response over a whole period is presented in Fig. One can recognize the drop in the heating power shortly after the voltage is applied.

This is due to the fact that also the heating resistor depends on temperature nonlinearity of the right hand side in 3.

An increase in temperature causes the heating resistance to grow which leads to slightly smaller heating power. This temperature is defined as the steady-state value. Thus, the temperature drops down to its initial value. FE mesh of the three-dimensional model with It considers the heat conduction through the solid material and the air beneath the membrane as well as convection to the air above the membrane.

The latter is considered in the form of convection boundary condition: Radiation mechanism has not been considered with respect to MOR, but it can be included via defining an additional temperature-dependant input as explained in section 5. Heat loss mechanisms considered by the FE model. System-level model of microstructure and electrical components. Resistors are modeled as temperature dependent. A step input function is applied to a controlled voltage source which drives the heating resistor. Two different applications are presented in the following sections. Simulation results of full scale FE model of silicon-nitride membrane from Fig.

Based on reduced model, it is possible to efficiently determine the control parameters while preserving the accuracy of the device simulation. The absolute temperature of the membrane should be independent of variations in ambient temperature and changes in convection boundary conditions. Furthermore, adjustments to new temperature set points should be performed by the thermal microsystem as quick as possible. In [18] two application scenarios for operation of the micro hotplate case study under temperature control were investigated, constant-value set-point temperature control and tracking control.

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In the first case a fast thermal response, which leads to the prescribed temperature value, is desired with minimum overshoot. The second scenario requires the temperature profile to track a prescribed function of time. Other books in this series. Piezoceramic Sensors Valeriy Sharapov. Silicon Microchannel Heat Sinks L.

Fast simulation of electro-thermal MEMS : efficient dynamic compact models in SearchWorks catalog

Micromechanical Photonics Hiroo Ukita. Laser Diode Microsystems Hans P.

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Modelling of Microfabrication Systems Raja Nassar. Shape Memory Microactuators Manfred Kohl. Capillary Forces in Microassembly Pierre Lambert. Micromachines as Tools for Nanotechnology Hiroyuki Fujita. Emphasis is placed on the application of the Arnoldi method for effective order reduction of thermal systems.