COMSOL 6.4 - Pull-In of an RF MEMS Switch

This model analyzes an RF MEMS switch consisting of a thin micromechanical bridge suspended over a dielectric layer. A DC voltage greater than the pull-in voltage is applied across the switch, causing the bridge to collapse onto the dielectric layer with a resulting increase in the capacitance of the device. A penalty-based contact force is implemented to model the contact forces as the bridge comes into contact with the dielectric. The capability of the multiphysics interface to handle both conducting and dielectric materials is also demonstrated.

Model Definition

Figure 1 shows the device geometry. The switch consists of a square polysilicon plate suspended 0.9 μm above a 0.1 μm thick thin film of silicon nitride (dielectric constant 7.5). Beneath the substrate is a silicon counter electrode that is grounded. The suspended plate is structurally anchored to the substrate by four rectangular flexures at its corners but is electrically isolated from the substrate.

Initially a small potential of 1 mV is applied to the polysilicon using the domain feature. This voltage is sufficient to measure the DC capacitance of the device. After 25 μs the applied voltage is increased by 5 V with a step function that has a rise time of 10 μs. The applied voltage is greater than the pull-in voltage of the structure and the switch pulls down onto the nitride. This process results in an abrupt and significant change in the capacitance of the device.

Due to the symmetry of the device, it is possible to model only a single quadrant of the structure.

The gap between the polysilicon and the nitride presents a model setup challenge. Since it is not possible to collapse the mesh to zero thickness (when the gap is closed) for numerical reasons, the mesh is compressed into the nitride layer as the structure deforms. The nitride layer itself is consequently not represented explicitly within the geometry, but instead is represented by means of a spatially varying function for the dielectric constant within the domain that contains both the nitride and the gap. Specifically, the dielectric constant in the domain is represented by a smoothed step function. The midpoint of the smoothed step function is chosen to be slightly above the height of the dielectric, so that when the polysilicon is in contact with the nitride, the dielectric constant in the domain takes the value of the nitride dielectric constant throughout the domain.

It is also worth noting that the multiphysics interface is capable of treating both conductors (for example, the polysilicon) and dielectric materials (for example, the nitride), as demonstrated in this model.

Figure 1: Top: Device geometry showing anchor points and symmetry planes. Bottom: Model geometry. Due to symmetry only one quadrant of the device needs to be modeled.

The contact between the polysilicon and the nitride is handled by an approximate penalty or barrier method, as described in Ref. 1. Stiff, nonlinear springs are used to represent the surface of the nitride. When the polysilicon is away from the nitride surface these springs have low stiffness and consequently have a negligible influence on the deformation. As the gap is reduced and approaches a predefined distance the spring becomes much stiffer and resists further closure. The contact forces Fc are given by:

where tn is the input estimate of the contact force, en is the penalty stiffness, g is the gap, that is, the distance between the polysilicon and the nitride.

Note that when this method is employed, it is important to tune the elastic stiffness and the contact force. The formula for the contact force is an approximate one and the model does not correctly reproduce the details of the dynamics of contact. However the model is primarily concerned with estimating the time the switch takes to make contact and with computing the initial and final capacitance of the switch.

For this same reason, a small mechanical damping is added to the Si domain to suppress the ringing of the structure once contact is made, thus preventing the time dependent solver from taking very small steps to resolve the ringing, which is not the focus of this model. (Of course care must be taken to ensure that the magnitude of the extra damping should not change the switching time constant significantly.)

Results and Discussion

Figure 2 shows the spatial dependence of the total displacement when the device is pulled in. Most of the structure is in contact with the nitride and the bending occurs primarily in the flexures and in the vicinity of their attachment points. The form of the contact pressure, shown in Figure 3, is consistent with this observation and it is interesting to note that the largest forces occur in the vicinity of the flexures.

Figure 2: Displacement of the polysilicon when pulled in. Most of the polysilicon structure is in contact with the silicon nitride with a displacement of 0.9 μm.

Figure 3: Contact forces acting on the polysilicon when the structure is pulled in.

Figure 4 shows the displacement of the switch as a function of time. The switch takes significantly longer to close than the time scale on which the applied voltage changes, primarily due to its inertia. Figure 5 shows the capacitance of the device as a function of time. The capacitance increases by a factor of approximately 65. Note that the capacitance changes on a significantly shorter time scale than the displacement.

Figure 4: Displacement of the center of the device as a function of time.

Figure 5: Capacitance of the device as a function of time. The transients in this plot that occur after the point of contact are not physical. The capacitance of the structure changes from 0.1 pF to 6.5 pF as a result of the pull-in.

Reference

1. M.A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures, volume 2: Advanced Topics, John Wiley & Sons Ltd., England, 1997.

MEMS_Module/Actuators/rf_mems_switch

Modeling Instructions

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Geometry 1

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Global Definitions

Parameters 1

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In the table, enter the following settings:


Name	Expression	Value	Description
V0	1[mV]	0.001 V	Initial voltage
Vstep	5[V]	5 V	Voltage step
insheight	100[nm]	1E-7 m	Insulator height
airheight	900[nm]	9E-7 m	Air height
en	1e15[Pa/m]	1E15 N/m³	Spring stiffness
tn	5e5[Pa]	5E5 Pa	Contact force

Definitions

Variables 1

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In the table, enter the following settings:


Name	Expression	Unit	Description
gap	airheight+w	m
contactpressure	(gap<=0)(tn-engap)+(gap>0)tnexp(-gap*en/tn)	Pa
Va	V0+Vstep*step2(t/1[s])

Geometry 1

Work Plane 1 (wp1)

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Work Plane 1 (wp1) > Plane Geometry

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Work Plane 1 (wp1) > Rectangle 1 (r1)

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Extrude 1 (ext1)

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In the table, enter the following settings:


Distances (µm)
1
2
4

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Form Union (fin)

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Definitions

Step 1 (step1)

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Step 2 (step2)

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Define selections.

Bridge

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Gap

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Bridge surface

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Base

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Box 3

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Bridge lower side

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Symmetry

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All domains

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Nonsolid

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Solid Mechanics (solid)

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Symmetry 1

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Boundary Load 1

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contactpressure	z

Fixed Constraint 1

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Linear Elastic Material 1

Add some small damping to reduce the transient effects at the final stage of the switch motion.

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Damping 1

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Electrostatics (es)

Charge Conservation in Solids 1

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Charge Conservation in Fluids 1

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Define the voltage applied to the bridge.

Domain Terminal 1

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Boundary Terminal 2

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Domain Terminal 1

Add another domain to the bridge. This terminal is used for the first () study step. It will be disabled in the transient step which will make active.

Domain Terminal 3