COMSOL 6.4 - Aluminum Nitride Lamb Wave Resonator

Aluminum Nitride Lamb Wave Resonator — Layered Shell Version

This tutorial demonstrates how to model an aluminum nitride Lamb wave resonator using the Electric Currents in Layered Shells interface in combination with the Layered Shell interface. The Lamb wave resonator (LWR) is an important building block in circuits used in mobile communication systems. Its operational characteristics are controlled by the piezoelectric material properties, the pattern of thin conducting layer on its surface, and its dimensions. Using the Piezoeletricity, Layered Shell multiphysics interface and materials from the Piezoelectric Material Library, you can create a digital prototype of an LWR.

This model is a layered shell version of Aluminum Nitride Lamb Wave Resonator — 3D, also in the MEMS Module Application Library, and requires the MEMS Module, Structural Mechanics Module, and Composite Materials Module.

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Introduction to the Composite Materials Module.

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Modeling Piezoelectricity: Which Module to Use?

Model Definition

In this tutorial, the LWR is an aluminum nitride (piezoelectric) block with an interdigitated electrode (IDE) on its surface, as shown in Figure 1. The base geometry used to set up the layered physics interfaces is shown in Figure 2. Suspended from four anchor points, the 0.4 μm piezoelectric block responds to AC excitation through the platinum IDE (80 nm). The LWR is designed to resonate at 7.9809 GHz.

The LWR geometric parameters are summarized in the first table in the Modeling Instructions section so the geometry can easily be modified for design optimization. The geometry is rebuilt automatically when geometric parameters are changed if the rules are followed. For example, n must be chosen so that (n+1)/4 is an integer, or n = {3, 7, 11, …}. Also, the overlap parameter op should be smaller than the length of the finger l, that is, op < l. Keep in mind, however, that changes in the geometry will shift the resonant frequency. An outline of the steps in the fabrication of an actual LWR can be found in Ref. 1.

In this model, the fully coupled structural and electrostatic equations are solved in the piezoelectric and conducting domain. The model also includes mechanical loss through an isotropic structural loss factor of 0.002 for the piezoelectric and the platinum layers.

In this tutorial we are interested in the symmetric S0 mode, which, in typical applications, results in maximal electromechanical coupling efficiency. Because of the symmetry of the device structure and the symmetry of the S0 mode, only one half of the device needs to be modeled. This reduces computational time, which is important because this model requires a fine mesh to compute accurate eigenmodes.

This tutorial shows the setup of Eigenfrequency and Frequency Domain studies. In the Eigenfrequency study you investigate the eigenmodes of the structure to find the S0 mode. In the subsequent Frequency Domain study, a 1 V drive signal is applied to the signal electrode and the resonator’s frequency response from 7.95 to 8.05 GHz is analyzed.

The results of this example is compared with the solid model variant Aluminum Nitride Lamb Wave Resonator — 3D. To match the results, the model setup needs to be the same. In the solid version of the example, the thin Platinum layer is not modeled explicitly in the Electrostatic interface, rather this thin layer modeled as a layer with either constant voltage or zero voltage using the and features. In the current example, the thin layer is considered as a conducting layer with uniform or zero voltage across all interfaces using the similar boundary conditions.

In the solid version of example, the Solid Mechanics interface has quadratic Serendipity discretization with three mesh elements across the thickness of the piezoelectric layer, and linear discretization with single mesh element across the conducting layer (through feature). While the Electrostatic interface has quadratic Lagrange discretization. In this example, the quadratic-linear discretization is chosen for the Layered Shell interface, and quadratic discretization for the Electric Currents in Layered Shells interface. Three mesh elements are used in the piezoelectric layer, while only one mesh element is used in the conducting layer. This is specified using the settings of nodes.

Figure 1: The 3D geometry of one half of the Lamb wave resonator.

Figure 2: The layered shell version of the model geometry.

Figure 3: Zones with different layers in the layered shell geometry. The red zone has only aluminum nitride, the green zone has aluminum nitride and platinum layers.

Figure 4: Cross-sectional view of zones having layers of different material and thickness.

Results and Discussion

Figure 5 shows the displacement in the z direction for the S0 mode at 7.9809 GHz. In high-frequency resonant devices such as LWRs, the Eigenfrequency study returns many extraneous solutions very close to the mode of interest. Only through visual inspection, one can distinguish the correct mode from the spurious modes. Figure 6 shows the solid displacement across the XZ cut plane through the center of the resonator to verify the S0 mode.

Figure 5: Displacement along the z-axis for the S0 mode at 7.7506 GHz.

Figure 6: Solid displacement across the XZ cut plane through the center of the resonator for the S0 mode at 7.9809 GHz.

Figure 7 shows a log plot of the admittance magnitude versus frequency from 7.95 to 8.05 GHz. The plot is annotated with the positions of resonance and anti-resonance peaks for the calculation of the electromechanical coupling coefficient.

Figure 7: Admittance versus frequency.

The frequency response yields important information relating to the device performance such as the effective electromechanical coupling coefficient given by

where fs and fp are the resonance and anti-resonance frequencies, respectively, with fs = 7.98, fp = 8.03, and kt2 = 1.526%. The effective electromechanical coupling coefficient is known to be a measure of the transduction efficiency for conversion of electrical into mechanical energy.

Notes About the COMSOL Implementation

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Modeling a composite laminated shell requires a 2D surface geometry, called a base surface, and a node that adds an extra dimension (1D) to the base surface geometry in the surface normal direction. Using the functionality, you can model several layers of different thicknesses, material properties, and fiber orientations. You can optionally specify the interface materials between the layers and the control mesh elements in each layer.

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The node is used to define various zones/sections of the Lamb wave resonator.

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The third direction for the selected coordinate system in the , , or represents the normal direction in the Layered Shell and Shell interfaces. This is also the direction in which the layer stacking is interpreted from bottom to top, and therefore, it is crucial to know it during modeling. There are two ways to achieve this:

Using physics symbols: Go to the physics settings, find the section, and select the checkbox. Then go to the material feature, for instance, , to see the normal direction represented by green arrows in the geometry.

Using result templates: When a solution dataset is available, use the result template to plot the normal direction.

Reference

1. J. Zou, C.M. Lin, A. Gao, and A.P. Pisano, “The Multi-Mode Resonance in AlN Lamb Wave Resonators,” J. Microelectromech. Syst., vol. 27, no. 6, pp. 973–84, 2018.

MEMS_Module/Piezoelectric_Devices/aln_lamb_wave_resonator_layered

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

Start by creating a new 3D model with a multiphysics interface.

In the tree, select > > > .

Click .

Click

In the tree, select > .

Click

Geometry 1

Use microns as the geometry unit.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Define and specify the parameters of the model.

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
tp	0.4[um]	4E-7 m	Thickness of piezoelectric layer
l	15[um]	1.5E-5 m	Length of finger
wf	0.2[um]	2E-7 m	Width of finger
op	14.6[um]	1.46E-5 m	Length of overlap, for op less than l
dy	(l-op)/2	2E-7 m	Electrode separation
n	11	11	Number of fingers, where (n+1)/4 = integer
we	0.2[um]	2E-7 m	Width of edge
la	0.4[um]	4E-7 m	Length of anchor
Vapp	1[V]	1 V	Applied voltage
eta0	2.0e-3	0.002	Loss factor for electrode layer
eta1	2.0e-3	0.002	Loss factor for piezoelectric layer
te	80[nm]	8E-8 m	Thickness of electrode layer

Add material models and layered materials under node.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > > .

Click the button in the window toolbar.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Global Definitions

Pt - Platinum (mat1)

In the window, under > click .

In the window for , click to expand the section.

In the tree, select > .

Click

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


Property	Variable	Value	Unit	Property group
Isotropic structural loss factor	eta_s	eta0	1	Basic

Locate the section. In the tree, select > .

Click

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


Property	Variable	Value	Unit	Property group
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	1	1	Basic

Layered Material: Platinum

In the window, right-click and choose .

In the window for , type Layered Material: Platinum in the text field.

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


Layer	Material	Rotation	Value	Thickness	Mesh elements
Layer 1	Pt - Platinum (mat1)	0.0	0 rad	te	1

Layered Material: Aluminum Nitride

Right-click and choose .

In the window for , type Layered Material: Aluminum Nitride in the text field.

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


Layer	Material	Rotation	Value	Thickness	Mesh elements
Layer 1	Aluminum Nitride (mat2)	0.0	0 rad	tp	3

Aluminum Nitride (mat2)

In the window, click .

In the window for , locate the section.

In the tree, select > .

Click

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


Property	Variable	Value	Unit	Property group
Isotropic structural loss factor	eta_s	eta1	1	Basic

Create the geometry model for half part of the lamb wave resonator.

Geometry 1

Work Plane 1 (wp1)

In the window, expand the > node.

Right-click and choose .

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

In the toolbar, click

In the window for , locate the section.

In the text field, type wf.

In the text field, type l.

Locate the section. From the list, choose .

In the text field, type dy/2.

Work Plane 1 (wp1) > Array 1 (arr1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

In the text field, type (n+1)/4.

Locate the section. In the text field, type 4*wf.

Work Plane 1 (wp1) > Move 1 (mov1)

In the toolbar, click

and choose .

Click the

button in the toolbar.

In the window for , locate the section.

Select the checkbox.

Locate the section. In the text field, type 2*wf.

In the text field, type -dy.

Work Plane 1 (wp1) > Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type (n-1/2)*wf+we.

In the text field, type 2*wf.

Locate the section. In the text field, type (l+dy)/2.

Work Plane 1 (wp1) > Rectangle 3 (r3)

In the toolbar, click

In the window for , locate the section.

In the text field, type 2*we.

In the text field, type la.

Locate the section. In the text field, type (n-1/2)*wf-we.

In the text field, type (l+dy)/2+2*wf.

Work Plane 1 (wp1) > Mirror 1 (mir1)

In the toolbar, click

and choose .

Select the objects and only.

In the window for , locate the section.

Select the checkbox.

Locate the section. In the text field, type 0.

In the text field, type -1.

Work Plane 1 (wp1) > Rectangle 4 (r4)

In the toolbar, click

In the window for , locate the section.

In the text field, type (n+1/2)*wf.

In the text field, type l+dy.

Locate the section. From the list, choose .

In the text field, type (n+1/2)*wf/2.

Work Plane 1 (wp1) > Partition Objects 1 (par1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Click to select the

toggle button for .

Select the object only.

Select the checkbox.

Work Plane 1 (wp1) > Delete Entities 1 (del1)

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

On the object , select Domain 1 only.

Click

In the window, right-click and choose .

Click the

button in the toolbar.

Define selections for the electrodes and other boundaries. This will make specifying the material models and physics interface settings easier.

Definitions

Signal

In the window, expand the > node.

Right-click and choose > .

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

Locate the section. From the list, choose .

Locate the section. In the text field, type -l/2.

In the text field, type 0.

Locate the section. From the list, choose .

Ground

Right-click and choose .

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

Locate the section. In the text field, type -Inf.

In the text field, type l/2.

Electrodes

In the toolbar, click

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

Locate the section. From the list, choose .

Locate the section. Under , click

In the dialog, in the list, choose and .

Click .

Symmetry

In the toolbar, click

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

Locate the section. From the list, choose .

Locate the section. In the text field, type 0.

Locate the section. From the list, choose .

Base Surface

In the toolbar, click

In the window for , type Base Surface in the text field.

Locate the section. From the list, choose .

Locate the section. In the text field, type 0.

Locate the section. From the list, choose .

Device

In the toolbar, click

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

Locate the section. From the list, choose .

Locate the section. In the text field, type -((l+dy)/2+2*wf+la)+0.01.

In the text field, type (l+dy)/2+2*wf+la-0.01.

Fixed Constraints

In the toolbar, click

In the window for , type Fixed Constraints in the text field.

Locate the section. From the list, choose .

Click

In the dialog, type 33 40 in the text field.

Click .

The stacking of the Platinum layer is not the same across the Aluminum Nitride base. In order to model the stacking, add a node with subnodes having different selections.

Materials

Layered Material Stack 1 (stlmat1)

In the window, under right-click and choose > .

In the window for , locate the section.

From the list, choose .

Aluminum Nitride

In the window, under > > click .

In the window for , type Aluminum Nitride in the text field.

Locate the section. From the list, choose .

Pt - Platinum

In the window, right-click and choose .

In the window for , type Pt - Platinum in the text field.

Locate the section. From the list, choose .

To visualize the stacking, create a plot through an action button in the section.

Layered Material Stack 1 (stlmat1)

In the window, click .

In the window for , click in the upper-right corner of the section. From the menu, choose .

Results

Layer Cross-Section Preview

In the window, expand the node, then click .

In the toolbar, click

Specify the settings for the interface.

Layered Shell (lshell)

Linear Elastic Material 1

In the window, under > click .

Damping 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Piezoelectric Material 1

In the window, under > click .

In the window for , locate the section.

Clear the checkbox.

In the table, clear the checkbox for .

Mechanical Damping 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Fixed Constraint 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Symmetry 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Continuity 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the window, click .

In the window for , click to expand the section.

From the list, choose .

Specify the settings for the interface.

Electric Currents in Layered Shells (ecis)

Conductive Shell 1

In the window, under > click .

Ground 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Conductive Shell 1

In the window, click .

Interface Terminal 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the V0 text field, type Vapp.

Piezoelectric Layer 1

In the window, under > click .

In the window for , locate the section.

Clear the checkbox.

In the table, clear the checkbox for .

Electric Continuity 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the table, enter the following settings:


	Layered material	Offset (m)
√	Aluminum Nitride	0

Create the mesh for the model.

Mesh 1

Free Quad 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Size 1

Right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section.

Select the checkbox. In the associated text field, type 0.1.

Click

Set up an study to search for an eigenfrequency around 8 GHz.

Eigenfrequency

In the window, click .

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

Locate the section. Clear the checkbox.

Step 1: Eigenfrequency

In the window, under click .

In the window for , locate the section.

Select the checkbox. In the associated text field, type 20.

From the list, choose .

In the text field, type 8.

Solution 1 (sol1)

In the toolbar, click

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Add a dataset and a dataset. This dataset will be used to plot the result of the study.

Results

Layered Material 1

In the window, expand the > node.

Right-click > and choose > .

Mirror 3D 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Click

With the solution from the study, create a 3D plot to display the shape of the eigenmode at 8 GHz. Use the dataset previously created.

Mode Shape

In the toolbar, click

In the window for , type Mode Shape in the text field.

Locate the section. From the list, choose .

Locate the section. Clear the checkbox.

From the list, choose .

Locate the section. Clear the checkbox.

Surface 1

Right-click and choose .

In the window for , locate the section.

In the text field, type w.

Select the checkbox.

Click to expand the section. Select the checkbox.

In the text field, type -1E10.

In the text field, type 1E10.

In the toolbar, click

Click the

button in the toolbar.

Click the

button in the toolbar.

Click the

button in the toolbar.

Click the

button in the toolbar.

With the solution from the study, create a plot to display the eigenmode in the xz cut plane through the center.

Mode Shape, Center XZ Plane

In the toolbar, click

In the window for , type Mode Shape, Center XZ Plane in the text field.

Locate the section. From the list, choose .

Locate the section. Clear the checkbox.

Slice 1

Right-click and choose .

In the window for , locate the section.

Select the checkbox.

Locate the section. From the list, choose .

In the text field, type 1.

Locate the section. From the list, choose .

In the text field, type 15.

Mode Shape, Center XZ Plane

In the window, click .

In the window for , locate the section.

Clear the checkbox.

In the toolbar, click

Click the

button in the toolbar.

Click the

button in the toolbar.

Set up a study with a range that includes the features of interest, for example, resonance and anti-resonance peaks. Disable the option to generate default plots from this study.

Add Study

In the toolbar, click