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Solidly Mounted Resonator 2D
Introduction
A solidly mounted resonator (SMR) is a micromachined, piezoelectric MEMS resonator formed on top of an acoustic mirror on a thick substrate. The acoustic mirror comprises alternating layers of materials with high and low acoustic impedances, which confines acoustic energy within the piezoelectric active layer. SMRs are very robust and have higher resonant frequencies than Bulk Acoustic Wave (BAW) resonators (in the GHz range). This tutorial shows how you can model SMR in 2D using eigenfrequency and frequency-response analyses.
Model Definition
The geometry of the SMR model and its key components are shown in Figure 1.
Figure 1: Model geometry showing the key components of the solidly mounted resonator.
Note that for clarity, the vertical scale is magnified to show the layers. All dimensions are parameterized in the model. Various selection features are used for the construction of the geometry and the setup of physics and mesh.
The fabrication of the device is discussed in Ref. 1. Here we provide a description of the final structure and an explanation of its principle of operation.
At the top of the device is a ZnO piezoelectric layer with aluminum electrodes at its top (drive) and bottom (ground) surfaces. Here, the pole direction is along the vertical axis and the piezoelectric material data is available in the built-in MEMS material library.
Underneath the piezoelectric resonator is a stack of alternating layers of molybdenum (high impedance) and silicon dioxide (low impedance). The thickness of the molybdenum and silicon dioxide layers were chosen to be 1.82 µm and 1.65 µm, respectively, to reflect the acoustic wave generated by the piezoelectric resonator and to prevent its dissipation in the silicon substrate. With this structure, the resonant frequency of the device is 870 MHz.
The parameters of the geometry are summarized in the first table in the section Modeling Instructions. The second table in the same section summarizes the material properties used in the model as specified in Ref. 1. Other material properties used in the model are obtained from the MEMS Module material library. As shown in the table, the Young’s moduli of the materials and the wavelength in silicon are computed from the values of density and acoustic velocity listed in the paper.
In this model, the fully coupled structural and electrostatic equations are solved in the piezoelectric layer, while only the structural equation is solved in other layers. Electrostatics equations are not solved in the aluminum layers because of its high electrical conductivity.
Perfectly Matching Layers (PML) boundary conditions are used at the sides and the bottom of the device to introduce anchor damping and eliminate reflections. The model also includes mechanical losses through an isotropic structural loss factor of 1.5 × 10-4. The model has fixed boundary conditions at the outer edges of the PML.
Note that although the structure has a plane of symmetry, which vertically bisects the device, we do not use a symmetry boundary condition because it would eliminate the anti-symmetric modes from the analysis.
This tutorial shows you the setup of Eigenfrequency and Frequency Domain studies. In the Eigenfrequency study you compute and investigate the eigenmodes of the structure. In the Frequency Domain study, 1 V drive signal is applied to the drive electrode and the frequency response of the resonator from 500 to 1200 MHz is analyzed.
To save time and reduce file size, a relatively coarse mesh is used, in particular in the horizontal direction. Thus only the main lower modes will be resolved in this model. The same approach was taken in the reference paper.
Figure 2: The mesh used in the model.
Results and Discussion
Figure 3 shows the mode shape of the fundamental mode of the resonator with the resonant frequency of about 870 MHz as intended by the design described in Ref. 1.
Figure 3: Surface plot of displacement indicating mode shape of the resonator’s fundamental mode.
Figure 4 is a surface plot of the displacement showing the excitation of the fundamental mode of the resonator when driven at the resonant frequency of about 870 MHz. The plot agrees reasonably well with Fig. 5(a) in Ref. 1.
Figure 4: Surface plot of displacement indicating the excitation of the resonator’s fundamental mode.
Figure 5 is a surface plot of the electric potential corresponding to Figure 4.
Figure 5: Surface plot of electric potential indicating the excitation of the resonator’s fundamental mode.
Figure 6 is a line plot of the displacement as a function of depth showing the confinement of acoustic energy within the piezoelectric layer, in agreement with Figure 5(b) in Ref. 1.
Figure 6: Displacement as function of depth showing the confinement of acoustic energy within the piezoelectric layer.
Figure 7 is a log plot of the magnitude of the impedance vs. frequency from 500 to 1200 MHz, in accordance with Figure 4 in Ref. 1.
Figure 7: |Z| versus frequency from 500 to 1200 MHz.
Reference
1. F.H. Villa-López and others, “Design and Modelling of Solidly Mounted Resonators for Low-Cost Particle Sensing,” Measurement Science and Technology, vol. 27, no. 2, 2016.
Application Library path: MEMS_Module/Piezoelectric_Devices/solidly_mounted_resonator_2d
Modeling Instructions
Start with a new 2D model with the built-in piezoelectric physics.
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select Structural Mechanics>Electromagnetics-Structure Interaction>Piezoelectricity>Piezoelectricity, Solid.
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Multiphysics>Eigenfrequency.
6
Click  Done.
Geometry 1
1
In the Geometry toolbar, click  Sketch.
Set the geometry unit to microns for convenience.
2
In the Model Builder window, click Geometry 1.
3
In the Settings window for Geometry, locate the Units section.
4
From the Length unit list, choose µm.
Since the device is much wider than its thickness, scale the graphics aspect ratio to zoom in in the thickness (Y) direction.
Definitions
In the Model Builder window, expand the Component 1 (comp1)>Definitions node.
Axis
1
In the Model Builder window, expand the Component 1 (comp1)>Definitions>View 1 node, then click Axis.
2
In the Settings window for Axis, locate the Axis section.
3
From the View scale list, choose Manual.
4
In the y scale text field, type 35.
5
Click  Update.
Enter geometry parameters. Note that we will truncate most of the thickness of the Si substrate and replace it with a perfectly matched layer (PML).
Global Definitions
Parameters 1 - Geometry
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, type Parameters 1 - Geometry in the Label text field.
3
Locate the Parameters section. In the table, enter the following settings:
Name
Expression
Value
Description
t_s
500[um]/25
2E-5 m
Substrate thickness (truncated)
t_i
200[nm]
2E-7 m
Insulator thickness
t_hil
1.82[um]
1.82E-6 m
High impedance layer thickness
t_lil
1.65[um]
1.65E-6 m
Low impedance layer thickness
t_pe
3.35[um]
3.35E-6 m
Piezoelectric layer thickness
t_e
200[nm]
2E-7 m
Electrode thickness
w_ar
200[um]
2E-4 m
Active area width
w_pe
800[um]
8E-4 m
Piezoelectric layer width
w
1000[um]
0.001 m
Device width
Enter material parameters. Then calculate the Young’s Modulus from the density and acoustic velocity for each linear material. Also calculate the wavelength in the substrate for an estimate of the PML thickness. A guessed value of 1.5e-4 is used for an isotropic damping factor to roughly match the vibration amplitude shown in the reference paper.
Parameters 2 - Material properties
1
In the Home toolbar, click  Parameters and choose Add>Parameters.
2
In the Settings window for Parameters, type Parameters 2 - Material properties in the Label text field.
3
Locate the Parameters section. In the table, enter the following settings:
Name
Expression
Value
Description
rho_ZnO
5680[kg/m^3]
5680 kg/m³
Density of ZnO
rho_Mo
10200[kg/m^3]
10200 kg/m³
Density of Mo
rho_SiO2
2170[kg/m^3]
2170 kg/m³
Density of SiO2
rho_Al
2700[kg/m^3]
2700 kg/m³
Density of Al
rho_Si
2330[kg/m^3]
2330 kg/m³
Density of Si
v_ZnO
6330[m/s]
6330 m/s
Acoustic velocity of ZnO
v_Mo
6280[m/s]
6280 m/s
Acoustic velocity_of Mo
v_SiO2
5540[m/s]
5540 m/s
Acoustic velocity of SiO2
v_Al
6450[m/s]
6450 m/s
Acoustic velocity of Al
v_Si
8320[m/s]
8320 m/s
Acoustic velocity of Si
E_Mo
rho_Mo*(v_Mo)^2
4.0227E11 Pa
Young's modulus of Mo
E_SiO2
rho_SiO2*(v_SiO2)^2
6.6601E10 Pa
Young's modulus of SiO2
E_Al
rho_Al*(v_Al)^2
1.1233E11 Pa
Young's modulus of Al
E_Si
rho_Si*(v_Si)^2
1.6129E11 Pa
Young's modulus of Si
eta0
1.5e-4
1.5E-4
Loss factor (guessed)
lambda_Si
v_Si/870[MHz]
9.5632E-6 m
Wavelength in Si
Build the parameterized geometry. Note how the selection and cumulative selection functionalities will be used to created named selections for material and physics settings later.
Geometry 1
Piezo - ZnO
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, type Piezo - ZnO in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type w_pe.
4
In the Height text field, type t_pe.
5
Locate the Position section. In the x text field, type -w_pe/2.
6
In the y text field, type t_e.
7
Locate the Selections of Resulting Entities section. Select the Resulting objects selection check box.
Top electrode - Al
1
Right-click Piezo - ZnO and choose Duplicate.
2
In the Settings window for Rectangle, type Top electrode - Al in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type w_ar.
4
In the Height text field, type t_e.
5
Locate the Position section. In the x text field, type -w_ar/2.
6
In the y text field, type t_pe+t_e.
7
Locate the Selections of Resulting Entities section. From the Show in physics list, choose All levels.
8
Find the Cumulative selection subsection. Click New.
9
In the New Cumulative Selection dialog box, type Al in the Name text field.
10
Click OK.
Bottom electrode - Al
1
Right-click Top electrode - Al and choose Duplicate.
2
In the Settings window for Rectangle, type Bottom electrode - Al in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type w_pe.
4
Locate the Position section. In the x text field, type -w_pe/2.
5
In the y text field, type 0.
Low impedance - SiO2
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, type Low impedance - SiO2 in the Label text field.
3
Locate the Size and Shape section. In the Width text field, type w.
4
In the Height text field, type t_lil.
5
Locate the Position section. In the x text field, type -w/2.
6
In the y text field, type -t_lil.
7
Click to expand the Layers section. In the table, enter the following settings:
Layer name
Thickness (µm)
Layer 1
(w-w_pe)/2
8
Select the Layers to the left check box.
9
Select the Layers to the right check box.
10
Clear the Layers on bottom check box.
11
Locate the Selections of Resulting Entities section. Find the Cumulative selection subsection. Click New.
12
In the New Cumulative Selection dialog box, type SiO2 in the Name text field.
13
Click OK.
Array - SiO2
1
In the Geometry toolbar, click  Transforms and choose Array.
2
In the Settings window for Array, type Array - SiO2 in the Label text field.
3
Locate the Input section. From the Input objects list, choose SiO2.
4
Locate the Size section. In the y size text field, type 3.
5
Locate the Displacement section. In the y text field, type -t_lil-t_hil.
High impedance - Mo
1
In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Low impedance - SiO2 (r4) and choose Duplicate.
2
In the Settings window for Rectangle, type High impedance - Mo in the Label text field.
3
Locate the Size and Shape section. In the Height text field, type t_hil.
4
Locate the Position section. In the y text field, type -t_lil-t_hil.
5
Locate the Selections of Resulting Entities section. Find the Cumulative selection subsection. Click New.
6
In the New Cumulative Selection dialog box, type Mo in the Name text field.
7
Click OK.
Array - Mo
1
In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Array - SiO2 (arr1) and choose Duplicate.
2
In the Settings window for Array, type Array - Mo in the Label text field.
3
Locate the Input section. Find the Input objects subsection. Click to select the  Activate Selection toggle button.
4
From the Input objects list, choose Mo.
5
Click  Build Selected.
Insulator - SiO2
1
In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Low impedance - SiO2 (r4) and choose Duplicate.
2
In the Settings window for Rectangle, type Insulator - SiO2 in the Label text field.
3
Locate the Size and Shape section. In the Height text field, type t_i.
4
Locate the Position section. In the y text field, type -(t_lil*3)-(t_hil*3)-t_i.
Substrate - Si
1
Right-click Insulator - SiO2 and choose Duplicate.
2
In the Settings window for Rectangle, type Substrate - Si in the Label text field.
3
Locate the Size and Shape section. In the Height text field, type t_s.
4
Locate the Position section. In the y text field, type -(t_lil*3)-(t_hil*3)-t_i-t_s.
5
Locate the Selections of Resulting Entities section. Find the Cumulative selection subsection. Click New.
6
In the New Cumulative Selection dialog box, type Si in the Name text field.
7
Click OK.
Bottom PML - Si
1
Right-click Substrate - Si and choose Duplicate.
2
In the Settings window for Rectangle, type Bottom PML - Si in the Label text field.
3
Locate the Size and Shape section. In the Height text field, type lambda_Si.
4
Locate the Position section. In the y text field, type -(t_lil*3)-(t_hil*3)-t_i-t_s-lambda_Si.
5
In the Geometry toolbar, click  Build All.
Create selections for the PML and fixed boundary condition.
Definitions
Not PML
1
In the Definitions toolbar, click  Box.
2
In the Settings window for Box, type Not PML in the Label text field.
3
Locate the Box Limits section. In the x minimum text field, type -w_ar.
4
In the x maximum text field, type w_ar.
5
In the y minimum text field, type -(t_hil*3)-(t_lil*3)-t_i-t_s/2.
PML
1
In the Definitions toolbar, click  Complement.
2
In the Settings window for Complement, type PML in the Label text field.
3
Locate the Input Entities section. Under Selections to invert, click  Add.
4
In the Add dialog box, select Not PML in the Selections to invert list.
5
Click OK.
Left fixed B.C.
1
In the Definitions toolbar, click  Box.
2
In the Settings window for Box, type Left fixed B.C. in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Boundary.
4
Locate the Box Limits section. In the x maximum text field, type -(w+w_pe)/4.
5
Locate the Output Entities section. From the Include entity if list, choose Entity inside box.
Right fixed B.C.
1
Right-click Left fixed B.C. and choose Duplicate.
2
In the Settings window for Box, type Right fixed B.C. in the Label text field.
3
Locate the Box Limits section. In the x minimum text field, type (w+w_pe)/4.
4
In the x maximum text field, type Inf.
Bottom fixed B.C.
1
In the Definitions toolbar, click  Explicit.
2
In the Settings window for Explicit, type Bottom fixed B.C. in the Label text field.
3
Locate the Input Entities section. From the Geometric entity level list, choose Boundary.
4
Select Boundaries 2, 21, and 49 only.
Fixed B.C.
1
In the Definitions toolbar, click  Union.
2
In the Settings window for Union, type Fixed B.C. in the Label text field.
3
Locate the Geometric Entity Level section. From the Level list, choose Boundary.
4
Locate the Input Entities section. Under Selections to add, click  Add.
5
In the Add dialog box, in the Selections to add list, choose Left fixed B.C., Right fixed B.C., and Bottom fixed B.C..
6
Click OK.
Create the PMLs.
Perfectly Matched Layer 1 (pml1)
1
In the Definitions toolbar, click  Perfectly Matched Layer.
2
In the Settings window for Perfectly Matched Layer, locate the Domain Selection section.
3
From the Selection list, choose PML.
Before adding material properties, set up the physics settings, so that the required properties will be highlighted when adding materials. In this 2D model, enter the out-of-plane thickness into each physics interface, so that lumped parameters can be computed appropriately. Use the selections made earlier for the physics selections. For Solid Mechanics: add damping subnodes and fixed boundary condition.
Solid Mechanics (solid)
1
In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).
2
In the Settings window for Solid Mechanics, locate the Thickness section.
3
In the d text field, type w_ar.
Linear Elastic Material 1
In the Model Builder window, under Component 1 (comp1)>Solid Mechanics (solid) click Linear Elastic Material 1.
Damping 1
1
In the Physics toolbar, click  Attributes and choose Damping.
2
In the Settings window for Damping, locate the Damping Settings section.
3
From the Damping type list, choose Isotropic loss factor.
Piezoelectric Material 1
1
In the Model Builder window, under Component 1 (comp1)>Solid Mechanics (solid) click Piezoelectric Material 1.
2
In the Settings window for Piezoelectric Material, locate the Domain Selection section.
3
From the Selection list, choose Piezo - ZnO.
Mechanical Damping 1
In the Physics toolbar, click  Attributes and choose Mechanical Damping.
Fixed Constraint 1
1
In the Physics toolbar, click  Boundaries and choose Fixed Constraint.
2
In the Settings window for Fixed Constraint, locate the Boundary Selection section.
3
From the Selection list, choose Fixed B.C..
For Electrostatics: only the domain surrounded by electrodes (the piezo domain) needs to be selected; remember to enter the out-of-plane thickness. Use the Terminal boundary condition (not the Electric Potential boundary condition) for the excitation port, so that lumped electrical parameters will be computed automatically.
Electrostatics (es)
1
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
2
In the Settings window for Electrostatics, locate the Domain Selection section.
3
From the Selection list, choose Piezo - ZnO.
4
Locate the Thickness section. In the d text field, type w_ar.
Ground 1
1
In the Physics toolbar, click  Boundaries and choose Ground.
2
In the Settings window for Ground, locate the Boundary Selection section.
3
From the Selection list, choose Bottom electrode - Al.
Terminal 1
1
In the Physics toolbar, click  Boundaries and choose Terminal.
2
In the Settings window for Terminal, locate the Boundary Selection section.
3
From the Selection list, choose Top electrode - Al.
4
Locate the Terminal section. From the Terminal type list, choose Voltage.
The domain and physics selections of the Piezoelectric Effect multiphysics coupling should be set up automatically.
Multiphysics
Piezoelectric Effect 1 (pze1)
Add material properties from the COMSOL Piezoelectric, MEMS, and Built-in material folders as an initial template. Then enter the available data from the reference paper using the parameters prepared earlier under Parameters 2 - Material properties.
Add Material
1
In the Home toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Piezoelectric>Zinc Oxide.
4
Click Add to Component in the window toolbar.
Materials
Zinc Oxide (mat1)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Piezo - ZnO.
3
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Density
rho
rho_ZnO
kg/m³
Basic
Loss factor for elasticity matrix cE
eta_cE_iso ; eta_cEii = eta_cE_iso, eta_cEij = 0
eta0
1
Stress-charge form
Add Material
1
Go to the Add Material window.
2
In the tree, select MEMS>Metals>Al - Aluminum.
3
Click Add to Component in the window toolbar.
Materials
Al - Aluminum (mat2)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Al.
3
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Isotropic structural loss factor
eta_s
eta0
1
Basic
Density
rho
rho_Al
kg/m³
Basic
Young’s modulus
E
E_Al
Pa
Young’s modulus and Poisson’s ratio
Add Material
1
Go to the Add Material window.
2
In the tree, select MEMS>Insulators>SiO2 - Silicon oxide.
3
Click Add to Component in the window toolbar.
Materials
SiO2 - Silicon oxide (mat3)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose SiO2.
3
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Isotropic structural loss factor
eta_s
eta0
1
Basic
Density
rho
rho_SiO2
kg/m³
Basic
Young’s modulus
E
E_SiO2
Pa
Young’s modulus and Poisson’s ratio
Add Material
1
Go to the Add Material window.
2
In the tree, select MEMS>Semiconductors>Si - Silicon (single-crystal, isotropic).
3
Click Add to Component in the window toolbar.
Materials
Si - Silicon (single-crystal, isotropic) (mat4)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Si.
3
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Isotropic structural loss factor
eta_s
eta0
1
Basic
Density
rho
rho_Si
kg/m³
Basic
Young’s modulus
E
E_Si
Pa
Young’s modulus and Poisson’s ratio
Add Material
1
Go to the Add Material window.
2
In the tree, select Built-in>Molybdenum.
3
Click Add to Component in the window toolbar.
4
In the Home toolbar, click  Add Material to close the Add Material window.
Materials
Molybdenum (mat5)
1
In the Settings window for Material, locate the Geometric Entity Selection section.
2
From the Selection list, choose Mo.
3
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Isotropic structural loss factor
eta_s
eta0
1
Basic
Density
rho
rho_Mo
kg/m³
Basic
Young’s modulus
E
E_Mo
Pa
Young’s modulus and Poisson’s ratio
To save time and file size, a relatively coarse mesh will be used, in particular in the horizontal direction. Thus only the main lower modes will be resolved in this model. The same approach was taken in the reference paper.
Mesh 1
Mapped 1
1
In the Mesh toolbar, click  Mapped.
2
In the Settings window for Mapped, locate the Domain Selection section.
3
From the Geometric entity level list, choose Domain.
4
From the Selection list, choose Top electrode - Al.
5
Click to expand the Control Entities section. Clear the Smooth across removed control entities check box.
6
Click to expand the Reduce Element Skewness section. Select the Adjust edge mesh check box.
Distribution 1
1
Right-click Mapped 1 and choose Distribution.
2
Select Boundary 45 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 16.
Distribution 2
1
In the Model Builder window, right-click Mapped 1 and choose Distribution.
2
Select Boundary 43 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 1.
Edge 1
1
In the Mesh toolbar, click  Edge.
2
Select Boundaries 42 and 47 only.
3
In the Settings window for Edge, click to expand the Control Entities section.
4
Clear the Smooth across removed control entities check box.
Distribution 1
1
Right-click Edge 1 and choose Distribution.
2
In the Settings window for Distribution, locate the Boundary Selection section.
3
In the list, select 42.
4
Click  Remove from Selection.
5
Select Boundary 47 only.
6
Locate the Distribution section. From the Distribution type list, choose Predefined.
7
In the Number of elements text field, type 16.
8
In the Element ratio text field, type 2.
9
From the Growth rate list, choose Exponential.
Distribution 2
1
Right-click Distribution 1 and choose Duplicate.
2
In the Settings window for Distribution, locate the Boundary Selection section.
3
Click  Clear Selection.
4
Select Boundary 42 only.
5
Locate the Distribution section. Select the Reverse direction check box.
Copy Edge 1
1
In the Model Builder window, right-click Mesh 1 and choose Copying Operations>Copy Edge.
2
Select Boundaries 42, 44, and 47 only.
3
In the Settings window for Copy Edge, locate the Destination Boundaries section.
4
Click to select the  Activate Selection toggle button.
5
Select Boundaries 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, and 41 only.
6
Click to clear the  Activate Selection toggle button.
Mapped 2
1
In the Mesh toolbar, click  Mapped.
2
In the Settings window for Mapped, locate the Control Entities section.
3
Clear the Smooth across removed control entities check box.
4
Locate the Reduce Element Skewness section. Select the Adjust edge mesh check box.
Distribution 1
1
Right-click Mapped 2 and choose Distribution.
2
Select Boundary 40 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 8.
Distribution 2
1
In the Model Builder window, right-click Mapped 2 and choose Distribution.
2
Select Boundaries 5 and 38 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 1.
Distribution 3
1
Right-click Mapped 2 and choose Distribution.
2
Select Boundaries 7, 9, 11, 13, 15, 17, 19, and 67 only.
Distribution 4
1
Right-click Mapped 2 and choose Distribution.
2
Select Boundaries 3, 22, 50, and 70 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
From the Distribution type list, choose Predefined.
5
In the Number of elements text field, type 16.
6
In the Element ratio text field, type 2.
7
From the Growth rate list, choose Exponential.
8
Select the Reverse direction check box.
Distribution 5
1
Right-click Mapped 2 and choose Distribution.
2
Select Boundary 1 only.
3
In the Settings window for Distribution, locate the Distribution section.
4
In the Number of elements text field, type 8.
5
Click  Build All.
Use the eigenfrequency study to look for the lower modes around 870 MHz.
Study 1 - Modes
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1 - Modes in the Label text field.
3
Locate the Study Settings section. Clear the Generate default plots check box.
Step 1: Eigenfrequency
1
In the Model Builder window, under Study 1 - Modes click Step 1: Eigenfrequency.
2
In the Settings window for Eigenfrequency, locate the Study Settings section.
3
Select the Desired number of eigenfrequencies check box. In the associated text field, type 4.
4
Find the Search region subsection. From the Unit list, choose MHz.
5
In the Search for eigenfrequencies around text field, type 870.
6
From the Eigenfrequency search method around shift list, choose Larger real part.
7
In the Home toolbar, click  Compute.
Create plot to visualize the mode shape. Then zoom in to the top part of the structure.
Results
Mode Shape (solid)
1
In the Home toolbar, click  Add Plot Group and choose 2D Plot Group.
2
In the Settings window for 2D Plot Group, type Mode Shape (solid) in the Label text field.
Surface 1
1
Right-click Mode Shape (solid) and choose Surface.
2
In the Settings window for Surface, locate the Coloring and Style section.
3
Click  Change Color Table.
4
In the Color Table dialog box, select Aurora>AuroraBorealis in the tree.
5
Click OK.
Deformation 1
1
Right-click Surface 1 and choose Deformation.
2
In the Settings window for Deformation, locate the Scale section.
3
Select the Scale factor check box. In the associated text field, type 2E12.
Materials
Molybdenum (mat5)
1
In the Model Builder window, under Component 1 (comp1)>Materials click Molybdenum (mat5).
2
In the Settings window for Material, locate the Geometric Entity Selection section.
3
Click  Zoom to Selection.
Results
Deformation 1
1
In the Model Builder window, under Results>Mode Shape (solid)>Surface 1 click Deformation 1.
2
In the Mode Shape (solid) toolbar, click  Plot.
The acoustic mirror effectively confines the mode energy at the top of the structure as expected.
Add a study for the frequency response. The frequency list is tailored to show more details near the main resonance and less elsewhere.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies>Frequency Domain.
4
Click Add Study in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
Step 1: Frequency Domain
1
In the Settings window for Frequency Domain, locate the Study Settings section.
2
From the Frequency unit list, choose MHz.
3
In the Frequencies text field, type range(500,50,800) range(810,10,850) range(860,2,870) range(870.1,0.05,870.9) range(871,1,910) 920 930 940 range(950,50,1200).
4
In the Model Builder window, click Study 2.
5
In the Settings window for Study, type Study 2 - Frequency Response in the Label text field.
6
Locate the Study Settings section. Clear the Generate default plots check box.
7
In the Home toolbar, click  Compute.
Create plot to visualize the electric potential solution at the main resonance.
Create plot to visualize the mode shape. Then zoom in to the top part of the structure.
Results
Electric Potential (es)
1
In the Home toolbar, click  Add Plot Group and choose 2D Plot Group.
2
In the Settings window for 2D Plot Group, type Electric Potential (es) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2 - Frequency Response/Solution 2 (sol2).
Surface 1
1
Right-click Electric Potential (es) and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type V.
4
Select the Description check box.
5
Locate the Coloring and Style section. Click  Change Color Table.
6
In the Color Table dialog box, select Rainbow>Dipole in the tree.
7
Click OK.
Electric Potential (es)
1
In the Model Builder window, click Electric Potential (es).
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Parameter value (freq (MHz)) list, choose 870.55.
4
In the Electric Potential (es) toolbar, click  Plot.
Add plots of the impedance and displacement to be compared with Fig.4 and 5 in the reference paper.
Log10|Z| - Fig.4
1
In the Home toolbar, click  Add Plot Group and choose 1D Plot Group.
2
In the Settings window for 1D Plot Group, type Log10|Z| - Fig.4 in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2 - Frequency Response/Solution 2 (sol2).
4
Click to expand the Title section. From the Title type list, choose Label.
5
Locate the Legend section. Clear the Show legends check box.
Global 1
1
Right-click Log10|Z| - Fig.4 and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
log10(abs(1/es.Y11)/1[ohm])
 
log10|Z| (Ohms)
4
In the Log10|Z| - Fig.4 toolbar, click  Plot.
Displacement profile - Fig.5(a)
1
In the Home toolbar, click  Add Plot Group and choose 2D Plot Group.
2
In the Settings window for 2D Plot Group, type Displacement profile - Fig.5(a) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 2 - Frequency Response/Solution 2 (sol2).
4
From the Parameter value (freq (MHz)) list, choose 870.55.
5
Click to expand the Title section. From the Title type list, choose Label.
Surface 1
1
Right-click Displacement profile - Fig.5(a) and choose Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type sqrt(abs(u)^2+abs(v)^2).
4
Click to expand the Range section. Select the Manual color range check box.
5
In the Maximum text field, type 0.02.
6
In the Displacement profile - Fig.5(a) toolbar, click  Plot.
Cut Line 2D 1
1
In the Results toolbar, click  Cut Line 2D.
2
In the Settings window for Cut Line 2D, locate the Data section.
3
From the Dataset list, choose Study 2 - Frequency Response/Solution 2 (sol2).
4
Locate the Line Data section. In row Point 1, set y to 4[um].
5
In row Point 2, set y to -20[um].
Y Displacement - Fig.5(b)
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Y Displacement - Fig.5(b) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Cut Line 2D 1.
4
From the Parameter selection (freq) list, choose From list.
5
In the Parameter values (freq (MHz)) list, select 870.55.
6
Locate the Title section. From the Title type list, choose Label.
7
Locate the Axis section. Select the Manual axis limits check box.
8
In the x minimum text field, type -0.02.
9
In the x maximum text field, type 0.02.
10
In the y minimum text field, type -20.5.
11
In the y maximum text field, type 4.5.
Line Graph 1
1
Right-click Y Displacement - Fig.5(b) and choose Line Graph.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type y.
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type -imag(v).
6
Select the Description check box. In the associated text field, type Displacement, Y component.
7
Click to expand the Coloring and Style section. From the Color list, choose Red.
8
From the Width list, choose 2.
Line Segments 1
1
In the Model Builder window, right-click Y Displacement - Fig.5(b) and choose Line Segments.
2
In the Settings window for Line Segments, locate the x-Coordinates section.
3
In the table, enter the following settings:
Expression
Unit
Description
-0.02
1
 
0.02
1
 
0.02
1
 
-0.02
1
 
-0.02
1
 
0.02
1
 
0.02
1
 
-0.02
1
 
-0.02
1
 
0.02
1
 
0.02
1
 
-0.02
1
 
-0.02
1
 
0.02
1
 
0.02
1
 
-0.02
1
 
-0.02
1
 
0.02
1
 
0.02
1
 
-0.02
1
 
-0.02
1
 
0.02
1
 
4
Locate the y-Coordinates section. In the table, enter the following settings:
Expression
Unit
Description
2*t_e+t_pe
µm
 
2*t_e+t_pe
µm
 
t_e+t_pe
µm
 
t_e+t_pe
µm
 
t_e
µm
Electrode thickness
t_e
µm
Electrode thickness
0
1
 
0
1
 
-t_lil
µm
 
-t_lil
µm
 
-t_lil-t_hil
µm
 
-t_lil-t_hil
µm
 
-2*t_lil-t_hil
µm
 
-2*t_lil-t_hil
µm
 
-2*t_lil-2*t_hil
µm
 
-2*t_lil-2*t_hil
µm
 
-3*t_lil-2*t_hil
µm
 
-3*t_lil-2*t_hil
µm
 
-3*t_lil-3*t_hil
µm
 
-3*t_lil-3*t_hil
µm
 
-3*t_lil-3*t_hil-t_i
µm
 
-3*t_lil-3*t_hil-t_i
µm
 
5
Click to expand the Coloring and Style section. From the Color list, choose Black.
Y Displacement - Fig.5(b)
In the Model Builder window, click Y Displacement - Fig.5(b).
Table Annotation 1
1
In the Y Displacement - Fig.5(b) toolbar, click  More Plots and choose Table Annotation.
2
In the Settings window for Table Annotation, locate the Data section.
3
From the Source list, choose Local table.
4
In the table, enter the following settings:
x-coordinate
y-coordinate
Annotation
-0.02
2
ZnO
-0.02
-0.85
SiO2
-0.02
-2.6
Mo
-0.02
-4.3
SiO2
-0.02
-6
Mo
-0.02
-7.8
SiO2
-0.02
-9.5
Mo
-0.02
-14
Substrate
5
Locate the Coloring and Style section. Clear the Show point check box.
6
From the Anchor point list, choose Middle left.
7
In the Y Displacement - Fig.5(b) toolbar, click  Plot.