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Multilayered microresonators commonly use sensitive coating or piezoelectric layers for detection of mass and gas. Most of these microresonators have a variable cross-section that complicates the prediction of their fundamental resonant frequency (generally of the bending mode) through conventional analytical models. In this paper, we present an analytical model to estimate the first resonant frequency and deflection curve of single-clamped multilayered microresonators with variable cross-section. The analytical model is obtained using the Rayleigh and Macaulay methods, as well as the Euler-Bernoulli beam theory. Our model is applied to two multilayered microresonators with piezoelectric excitation reported in the literature. Both microresonators are composed by layers of seven different materials. The results of our analytical model agree very well with those obtained from finite element models (FEMs) and experimental data. Our analytical model can be used to determine the suitable dimensions of the microresonator’s layers in order to obtain a microresonator that operates at a resonant frequency necessary for a particular application.

Multilayered microresonators fabricated using microelectromechanical systems (MEMS) have potential applications such as mixer-filters [

The microresonators are extremely sensitive to surface processes due to their large surface area to mass ratios [

An important issue in the mechanical design process of a multilayered microresonator with variable cross-section is the correct determination of its fundamental resonant frequency. Another important parameter is the estimation of its deflection under an excitation load, which can be used to study its mechanical behavior. Theoretical models can predict the resonant frequency shift of microresonators caused by variations of their geometrical variables and mechanical properties. Lobontiu

This paper is organized as follows: in Section 2, we describe the analytical model for the bending resonant frequency of a multilayered microresonator with variable cross section. Next, in Section 3, this model is applied to two multilayered microresonators with mass sensing applications reported in the literature. The results of our analytical model are compared with those obtained from a simple cantilever model, FEMs, and experimental data. Finally, we presented our conclusions and proposed future research in Section 4.

This section describes the analytical modeling to predict the first bending resonant frequency and deflection of multilayered microresonators with variable cross section. Rayleigh’s method can predict the first bending resonant frequency of structures with complex geometrical shapes considering energy conservation in the structures [

The bending deflection response

For out-of plane bending in single and double clamped beams with length _{m}_{m}

Based on Rayleigh’s method, the first bending resonant frequency (_{r}_{m}_{m}_{m}_{m}_{r}

In order to use the Rayleigh’s method in a multilayered microresonator with variable cross section, we need to know its elastic centroid, bending stiffness and mass per unit length. We propose a multilayered microresonator with variable cross section, as shown in _{iSj}_{iSj}_{iSj}

For this case, _{0} and M_{0} represent the total reaction load and bending moment at the fixed end of the microresonator. Furthermore, _{Sj}

Multilayered microresonator is integrated by layers made of homogeneous and isotropic materials. The film mechanical properties of the different layers must be considered;

The microresonator layers are symmetrically distributed with respect to

The bending vibration of the microresonator occurs along of the

The plane sections of layers do not deform,

The

The total length of the microresonator is greater than 10 times its total thickness. This is a necessary condition to use the Euler-Bernoulli beam theory;

Damping and surface effects (e.g., surface energy, surface tension, surface relaxation, and surface reconstruction) are neglected;

The residual stress in the microresonator is neglected;

The nonlineality and viscosity of the layers are not considered;

The undercut to the microresonator caused by etching processes is neglected.

For our microresonator, the elastic centroid _{Sj}_{iSj}_{iSj}_{(i−1)Sj}, the subscript _{j}_{Sj}_{Sj}_{iSj}_{(i−1)Sj} is distance from the bottom plane of the first layer to the top plane of the (_{Sj}_{Sj}_{iS1} considers the overall width of the _{iS1} includes the sum of the width of the three layers located on the same distance _{iS1}_{0Sj}

The effective bending stiffness _{z}_{Sj}

The maximum potential energy (_{mL}_{mL}_{S12} = _{S1} + _{S2}, _{S123} = _{S1} + _{S2} + _{S3}, _{S1234} = _{S1} + _{S2} + _{S3}+ _{S4}, and

Using the Rayleigh’s method, the first bending resonant frequency of the multilayered microresonator is obtained as:

_{add}_{add}_{add}y^{2}(_{add}

The deflection _{Sj}^{ε}^{ε}

Thus, the deflection in each one of the four sections of proposed microresonator are determined from:
_{Sj}

The deflection _{Sj}

Using Macaulay’s method, we find the total load function _{Sj}_{iS1} considers the overall width of the all layers located in _{iS1} (e.g., if there are three layers made with the same material located to a distance _{iS1}, then _{iS1} includes the sum of each width of the layers).

The shear load function

Integrating _{1} and _{2} are constants calculated through boundary conditions (_{0} and _{0}) of the shear load and bending moment functions. Substituting these conditions into _{1} = _{2} = 0.

The bending moment functions for each one of the four sections of the multilayered microresonator are determined from

For 0 < _{S1}
_{S1} < _{S12}
_{S12} < _{S123}
_{S123} < _{S1234}

Then, the deflection _{Sj}_{Sj}

For 0 < _{S1}
_{S1} < _{S12}
_{S12} < _{S123}
_{S123} < _{S1234}

For multilayered microresonators with two or three different sections, only the first two or three deflections equations must be used. Next, _{Sj}

We applied our analytical model to estimate the first bending resonant frequency and deflection of two multilayered microresonators reported in the literature. In addition, we used finite element models (FEMs) to study the bending vibration of these microresonators. Our analytical results agree with the FEMs and experimental results of the two microresonators.

We considered the geometrical configuration of two multilayered microresonators for mass sensing applications developed by Lu

These microresonators have self-actuation and self-sensing capability. The details of their fabrication process can be found elsewhere [_{2}), titanium (Ti), platinum (Pt), PZT, chromium (Cr), and gold (Au). The order of the nine layers on the first and second section is the following: Si/SiO_{2}/Ti/Pt/PZT/Cr/Au/Cr/SiO_{2}. Only the first section has two holes to decrease initial stress due to PZT deformation, which improve the signal to noise of the piezoresistive gauge contained in the first section. The third only contains a Si layer and the fourth section has three layers of Si/Cr/Au.

_{iS1} include the overall width of the layers located to a same distance _{iS1}.

_{2}) and the third section is only formed by Si (see

Both microresonator type-A and type-B have four sections with different numbers of layers in each one of them. In addition, their layers have a symmetrical configuration with respect to the _{r}

For a simple cantilever model where the origin of its coordinate system is located at its fixed end and the _{rc}_{c}_{c}_{c}_{c}

The deflection curve (_{c}_{c}_{c}

Both microresonators type-A and type-B can be approximated to a simple cantilever model if only a layer in each one of their four sections is considered. For this, we used the layer with the largest thickness, which corresponds to the silicon layer for both microresonators. For the microresonator type-A, we neglected the holes and considered a silicon layer with dimensions 700 × 150 × 5 μm. This layer presents dimensions much larger than any other layer of the microresonator type-A. For the microresonator type-B, the dimensions (80 × 38 × 1 μm) of its piezoelectric layer (PZT) are close to those of its silicon layer (200 × 50 × 5.25 μm). Using

In addition, we made two FEMs for both microresonators through ANSYS^{®} software. These models used solid95 type elements, in which each element is defined by 20 nodes with three degrees of freedom per node: translations in the nodal _{2}, and PZT) in order to obtain a mesh that does not overcome the maximum nodes number (125,000 nodes) allowed by our ANSYS^{®} software license.

The thinner layers of the microresonator type-A have thicknesses less than 300 nm, which significantly increases the number of nodes in the FEM mesh. This is a problem to mesh layers on the order of nanometers. Similarly, the FEM of the microresonator type-B only included the thicker layers (silicon, SiO_{2}, and PZT layers).

In addition, we compared the normalized deflection _{max}

Our analytical model can be used in the design phase of multilayered microresonators with variable cross section in order to estimate their lowest bending resonant frequencies and deflections. Using our model, a designer can determine the dimensions of the microresonator’s layers that allow it to operate at a resonant frequency suitable to a particular application. A designer can also use the proposed model to know the influence of the microresonator’s materials on its resonant frequency.

An analytical model to estimate the first bending resonant frequency and deflection curve of multilayered microresonator with variable cross-section was presented. This model is formulated through the Rayleigh and Macaulay methods, as well as the Euler-Bernoulli beam theory. We have applied our analytical model to two multilayered microresonators composed by layers of seven different materials reported in the literature. The results of proposed analytical model presented a relative difference less than 3.0% with respect to experimental data. Our analytical model can be useful in the mechanical design of multilayered microresonator for detection of mass and chemical species. Future work will include the effect of the residual stress on the multilayered microresonators with variable cross-section. In addition, we will study the resonant frequency shift of these microresonators due to films or particles deposited on their surface.

This work was supported by the Mexican National Council for Science and Technology (CONACYT) through grant 84605.

_{3}Al films

View of the multilayered microresonator with variable cross section proposed in this work.

Geometrical nomenclature proposed for the

Examples of multilayered microresonators obtained from the proposed multilayered microresonator. These microresonators could are composed by silicon layers with (

View of the load types acting on the proposed multilayered microresonator.

SEM micrograph of the multilayered microresonator type-A [

SEM micrograph of the multilayered microresonator type-B [

Geometrical configuration of the layers and load types on the microresonator type-A. This figure is not drawn in scale.

Detail view of the layers located on the first three sections of the microresonator type-A. This figure is not drawn in scale.

Detail view of the layers located on the first three sections of the microresonator type-A. This figure is not drawn in scale.

Geometrical configuration of the layers and load types on the microresonator type-B. This figure is not drawn in scale.

Detail view of the layers located on the first three sections of the microresonator type-B. This figure is not drawn in scale.

First bending mode of the microresonator type-A obtained using a FEM.

First bending mode of the microresonator type-B obtained using a FEM.

Normalized deflection _{max} of the microresonator type-A obtained through our analytical model, a FEM, and a simple cantilever model.

Normalized deflection _{max} of the microresonator type-B obtained through our analytical model, a FEM, and a simple cantilever model.

Order of the layers in the four sections of the microresonator type-A.

| ||||
---|---|---|---|---|

Si | 1st | 1st | 1st | 1st |

SiO_{2} |
2nd | 2nd | ||

Ti | 3rd | 3rd | ||

Pt | 4th | 4th | ||

PZT | 5th | 5th | ||

Cr | 6th | 6th | 2nd | |

Au | 7th | 7th | 3rd | |

Cr | 8th | 8th | ||

SiO_{2} |
9th | 9th |

Geometrical parameters of the layers used in the microresonator type-A.

| |||
---|---|---|---|

_{S1} = _{S3} |
50 | _{2S3} = _{3S4} |
130 |

_{S2} |
100 | _{1S1} = _{1S2} = _{1S3} = _{1S4} |
5.0 |

_{S4} |
500 | _{2S1} = _{2S2} |
0.40 |

_{1S1} |
140 | _{3S1} = _{3S2} |
0.05 |

_{2S1} |
120 | _{4S1} = _{4S2} |
0.25 |

_{3S1} = _{4S1} = _{5S1} = _{6S1} |
72 | _{5S1} = _{5S2} |
1.0 |

_{7S1} = _{8S1} = _{9S1} = _{3S2} |
72 | _{6S1} = _{6S2} = _{8S1} = _{8S2} |
0.03 |

_{4S2} = _{5S2} = _{6S2} = _{7S2} |
72 | _{7S1} = _{7S2} = _{3S4} |
0.15 |

_{8S2} = _{9S2} |
72 | _{9S1} = _{9S2} |
0.30 |

_{1S3} = _{1S4} |
150 |

Order of the layers in the four sections of the microresonator type-B.

| ||||
---|---|---|---|---|

Si | 1st | 1st | 1st | 1st |

SiO_{2} |
2nd | 2nd | ||

Ti | 3rd | |||

Pt | 4th | |||

PZT | 5th | |||

Cr | 6th | 2nd | ||

Au | 7th | 3rd | ||

Cr | 8th | |||

SiO_{2} |
9th |

Geometrical parameters of the layers used in the microresonator type-B.

| |||
---|---|---|---|

_{S1} |
80 | _{1S1} = _{1S2} = _{1S3} = _{1S4} |
5.25 |

_{S2} |
6 | _{2S1} = _{2S2} |
0.40 |

_{S3} |
14 | _{3S1} |
0.05 |

_{S4} |
100 | _{4S1} |
0.25 |

_{1S1} = _{2S1} = _{1S2} = _{2S2} |
50 | _{5S1} |
1.0 |

_{3S1} = _{4S1} = _{5S1} = _{6S1} |
38 | _{6S1} = _{8S1} = _{2S4} |
0.03 |

_{7S1} = _{8S1} = _{9S1} |
38 | _{7S1} |
0.15 |

_{2S4} = _{3S4} |
44 | _{9S1} |
0.30 |

_{1S3} = _{1S4} |
50 | _{3S4} |
0.24 |

Mechanical properties of the layers used in the microresonator type-A and microresonator type-B.

^{−3}) |
|||
---|---|---|---|

| |||

Si | 130 | 2330 | 0.28 |

SiO_{2} |
73 | 2200 | 0.17 |

Ti | 110 | 4510 | 0.32 |

Pt | 168 | 21,400 | 0.39 |

PZT | 63 | 7500 | 0.36 |

Cr | 140 | 7190 | 0.21 |

Au | 75 | 19,300 | 0.42 |

Values of the weight by unit length, reaction load, bending moment, and effective stiffness for the microresonator type-A and microresonator type-B.

| ||
---|---|---|

_{0} × 10^{9} N |
15.90 | 1.96 |

_{0} × 10^{12} N·m |
5.24 | 0.17 |

_{S1} × 10^{6} N·m^{−1} |
29.09 | 12.79 |

_{S2} × 10^{6} N·m^{−1} |
30.32 | 6.43 |

_{S3} × 10^{6} N·m^{−1} |
17.14 | 6.00 |

_{S4} × 10^{6} N·m^{−1} |
21.11 | 8.09 |

(_{z}_{S1} × 10^{12} N·m^{2} |
344.98 | 159.86 |

(_{z}_{S2} × 10^{12} N·m^{2} |
361.38 | 89.57 |

(_{z}_{S3} × 10^{12} N·m^{2} |
203.12 | 78.38 |

(_{z}_{S4} × 10^{12} N·m^{2} |
216.24 | 85.56 |