Disc Resonator Anchor Losses

Electromechanical resonators and filters are widely used in signal processing and communications applications. Over the past decade significant progress has been made in miniaturizing these systems using microfabrication technologies. In many of these applications it is important to produce resonators with high quality factors. Several mechanisms limit the obtainable quality factors of MEMS resonators including thermoelastic damping (for flexural modes), material losses (the fundamental limiting factor on the obtainable quality factor), and anchor losses. Anchor losses occur when the vibration of the structure and its supporting anchors excites acoustic waves propagating in the substrate. These waves are radiated away from the resonator resulting in a loss of mechanical energy, or damping. In many cases anchor damping can represent the limiting loss mechanism that determines the resonator quality factor. Modifications in the anchor design can significantly effect the coupling of the resonant mode to the substrate, so numerical simulations of anchor damping can be a useful tool to employ as part of the resonator design process.

This model shows how to determine the anchor damping limited resonator quality factor of a diamond disc resonator, using a PML to absorb the waves propagating in the substrate. It is based on the work presented in Ref. 1 and Ref. 2.

Model Definition

Figure 1 shows a slice through the structure on a plane with an edge parallel to the line of axial symmetry. The critical components of the device are highlighted. The resonator is attached to the substrate by a polysilicon stem. The disc (polycrystalline diamond) and stem (polysilicon) are made from different materials so that there is an impedance mismatch between the materials. This mismatch is designed to suppress energy transfer to the substrate, enhancing the quality factor of the device Ref. 1. A hemispherical section of the substrate is modeled and the waves propagating away from the anchor are absorbed by a perfectly matched layer (PML), which is applied to a layer at the edge of the substrate. The outer surface of the PML is fixed.

Since the fundamental model has axial symmetry, the device is modeled using a 2D axisymmetric geometry.

Figure 1: Axisymmetric model geometry, with key parts of the structure highlighted.

Results and Discussion

Figure 2 shows the resonator displacement in the fundamental mode of the resonator, which occurs at a frequency of 537 MHz. This compares reasonably well with the experimentally measured value of the resonant frequency, which was 546 MHz. Figure 3 shows a 3 dimensional rendering of the displacement, with a color scale chosen to emphasize the waves propagating in the substrate. The energy transmitted by these waves is responsible for the anchor losses. The effect of the PML is also apparent in this figure, and it seems that the waves are absorbed well by the PML in this model.

The quality factor of the resonator is related to the ratio of the real and imaginary parts of the eigenvalue of the mode. In this case the computed quality factor is 22,000, which is a similar magnitude to the quality factor measured in vacuum of approximately 17,500 (Ref. 1). This indicates that anchor damping is likely to be the dominant loss mechanism for devices designed in this manner.

Figure 2: Section through the axisymmetric vibration mode of the disc showing the displacement of the structure. The disc undergoes uniform extension along the radial direction.

Figure 3: Total displacement of the structure as a whole with a color scale chosen to emphasize the waves propagating in the substrate.

References

1. J. Wang, J.E. Butler, T. Feygelson, and C. T.-C. Nguyen, “1.51-GHz nanochrystalline diamond micromechanical disk resonator with material-mismatched isolating support,” Proceedings of the 17th International IEEE Microelectromechanical Systems Conference, pp. 641–644, Maastricht, 2004.

2. P. Steeneken “Parameter Extraction and Support-Loss in MEMS Resonators”, COMSOL Conference 2007, Grenoble. (Available at: https://www.comsol.com/paper/download/43866/Steeneken_pres.pdf and https://arxiv.org/ftp/arxiv/papers/1304/1304.7953.pdf.)

MEMS_Module/Actuators/disc_resonator_anchor_losses

Modeling Instructions

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Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
R_d	10[um]	1E-5 m	Disc radius
H_d	3[um]	3E-6 m	Disc height
H_p	0.8[um]	8E-7 m	Post height
R_p	0.8[um]	8E-7 m	Post radius
R_s	40[um]	4E-5 m	Substrate section radius
t_PML	wl	9.5E-6 m	PML thickness
wl	9.5[um]	9.5E-6 m	Wavelength of acoustic waves

Geometry 1

Rectangle 1 (r1)

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the text field, type R_d.

In the text field, type H_d.

Locate the section. In the text field, type H_p.

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type R_p.

In the text field, type 2*H_d+2*H_p.

Rectangle 3 (r3)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2.5*R_p.

In the text field, type H_d.

Locate the section. In the text field, type H_d+2*H_p.

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type R_s+t_PML.

In the text field, type 90.

Locate the section. In the text field, type 270.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	t_PML
Layer 2	R_s+-R_d

Click

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button in the toolbar.

Definitions

Perfectly Matched Layer 1 (pml1)

In the toolbar, click

Select Domain 1 only.

Materials

Silicon Substrate

In the window, under right-click and choose .

In the window for , type Silicon Substrate in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	130[GPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.28	1	Young’s modulus and Poisson’s ratio
Density	rho	2230[kg/m^3]	kg/m³	Basic

Polysilicon

Right-click and choose .

In the window for , type Polysilicon in the text field.

Select Domains 4–7 and 9 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	150[GPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.22	1	Young’s modulus and Poisson’s ratio
Density	rho	2330[kg/m^3]	kg/m³	Basic

Diamond

Right-click and choose .

In the window for , type Diamond in the text field.

Select Domain 8 only.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	1061[GPa]	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.12	1	Young’s modulus and Poisson’s ratio
Density	rho	3440[kg/m^3]	kg/m³	Basic