An imaging method for the eigenmode shape of FBAR based on the semi-analytical finite element theory

Abstract

The present invention provides an imaging method for the singular mode vibration mode of FBAR based on the semi - analytical finite element theory, including the following steps: Step S1, constructing a semi - analytical finite element formulation of a FBAR with a complex electrode configuration; Step S2, using triangular elements to mesh the upper surface of the FBAR to obtain the coordinates of three nodes in each mesh element; Step S3, determining the interpolation shape function, stiffness matrix and mass matrix of each mesh element according to the node coordinates; Based on the semi - analytical finite element theory, the present invention characterizes the vibration mode of a three - dimensional FBAR at ultra - high frequencies, reduces the three - dimensional modal vibration imaging of a three - dimensional FBAR device to two - dimensional planar imaging. The simplification of the element type and the substantial reduction in the number of elements make the imaging technology provided by the present invention have a lower time cost. At the same time, it does not require various parameter settings and post - processing in commercial finite element software, realizing fast and quantitative imaging of the FBAR vibration mode.

Description

Technical Field

[0001] The present invention relates to the field of computer technology, and more particularly, to an imaging method for the singular mode shape of an FBAR based on the semi-analytical finite element theory. Background Art

[0002] A film bulk acoustic resonator (FBAR) is a new type of electronic component that provides an accurate frequency reference. It has outstanding advantages such as micron-scale size, ultra-high frequency in the GHz range, and silicon-based chip integration. It is the key physical basis for constructing core electronic devices such as filters and system architectures, and has been widely used in key fields such as third-generation wireless communication. The core structure of an FBAR is a piezoelectric oscillation stack structure composed of an upper electrode, a piezoelectric thin film, a lower electrode, and a silicon substrate. By applying an alternating voltage between the upper and lower electrodes, the piezoelectric oscillation stack is driven to resonate at an ultra-high frequency.

[0003] To meet different performance requirements, the upper drive electrodes of FBARs will adopt different geometric configurations, such as rectangles, circles, and complex irregular pentagons. These electrode configurations with different shapes will cause the FBAR to exhibit singular mode shapes, and quantifying the imaging of these singular mode shapes is a difficult problem in the research of FBAR technology.

[0004] To address the problem of imaging the mode shape of an FBAR at ultra-high frequencies, researchers have developed some technical methods based on experimental measurements and commercial finite element software. For example, Shao Lei et al. from Shanghai Jiao Tong University measured the mode shape of an FBAR with an irregular pentagonal electrode through femtosecond laser testing technology, achieving effective imaging of the singular mode shape at ultra-high frequencies; Wu Biyan simulated the mode shapes of FBARs with different drive electrode configurations, including square, circular, and regular pentagonal electrodes, using ANSYS finite element software.

[0005] However, the existing technical methods have some limitations in the modal vibration mode imaging of FBAR at ultra-high frequencies, including: 1) In terms of experiments, the modal vibration mode imaging based on femtosecond laser testing technology requires a femtosecond-level three-dimensional laser vibrometer and a high frequency resolution, which have very high requirements for experimental equipment and testing, and the corresponding testing, processing, and time costs are very large, making it not suitable for batch imaging of the modal vibration modes of FBAR devices; 2) The FBAR three-dimensional simulation technology based on commercial finite element software often depends on the quality and large quantity of mesh division. As a result, the time cost required for three-dimensional FBAR simulation is large, and it is necessary to manually screen the actual vibration modes from the simulation results, and the post-processing is very cumbersome; 3) Due to the large economic and time costs of the existing technology, it is difficult to perform batch imaging of the modal vibration modes of FBAR, so it is difficult to be used in the engineering design and optimization of FBAR device structures. In summary, there is an urgent need to develop fast and quantitative imaging technology to solve the problem of difficult characterization of singular modal vibration modes of three-dimensional FBAR at ultra-high frequencies. Summary of the Invention

[0006] In order to solve the technical problems existing in the above background technology, the present invention provides an imaging method for the singular modal vibration modes of FBAR based on the semi-analytical finite element theory.

[0007] The present invention provides an imaging method for the singular modal vibration modes of FBAR based on the semi-analytical finite element theory, including the following steps:

[0008] Step S1, construct a semi-analytical finite element formulation of FBAR with a complex electrode configuration;

[0009] Step S2, use triangular elements to divide the mesh of the upper surface of FBAR to obtain the coordinates of three nodes in each mesh element;

[0010] Step S3, determine the interpolation shape function, stiffness matrix, and mass matrix of each mesh element according to the node coordinates;

[0011] Step S4, assemble the total stiffness matrix and total mass matrix based on the stiffness matrix and mass matrix of each mesh element, and solve to obtain the resonant frequency of FBAR;

[0012] Step S5, calculate the displacement vectors of all nodes in each mesh element according to the resonant frequency of FBAR, and combine the interpolation shape function of each mesh element to obtain the thickness stretching modal displacement of each node of FBAR, and draw the modal vibration mode corresponding to the resonant frequency.

[0013] In some embodiments, in step S1, it specifically includes:

[0014] For the area A1 on the upper surface of FBAR plated with a driving electrode, the two-dimensional scalar equation is:

[0015]

[0016] For region A2 on the upper surface of the FBAR where there is no driving electrode deposited, the two-dimensional scalar equation is

[0017]

[0018] where M n represents the coefficient of the two-dimensional scalar equation corresponding to the nth-order thickness-stretch mode. When n = 1, it represents the first-order thickness-stretch mode; f n represents the displacement component of the nth-order thickness-stretch mode along the coordinate component z in the thickness direction, and x and y respectively represent the coordinate components in two in-plane directions, ρ f represents the mass density of the piezoelectric thin film in the FBAR, ω represents the resonance frequency of the nth-order thickness-stretch mode of the FBAR, and ω n respectively represent the resonance frequency of the nth-order thickness-stretch mode of an infinitely large FBAR completely covered with the upper electrode and the resonance frequency of the nth-order thickness-stretch mode of an infinitely large FBAR completely not covered with the upper electrode.

[0019] In some embodiments, in step S2, it further includes:

[0020] Construct the variational functional of the overall FBAR structure, expressed as:

[0021]

[0022] Divide the cross-sections of region A1 and region A2 into n1 and n2 elements respectively, then the total energy of all elements is expressed as:

[0023]

[0024] where

[0025]

[0026] In some embodiments, in step S2, it further includes:

[0027] The nodal displacement vector on the planar element divided on region A1 is denoted as The interpolation shape function within the planar element is expressed as The thickness-stretch mode displacement function at any point within the planar element is expressed as:

[0028]

[0029] The energy functional of the planar element is expressed as:

[0030]

[0031] Among them,

[0032]

[0033] and are the stiffness matrix and mass matrix of the plane element .

[0034] In some embodiments, in step S2, it further includes:

[0035] The nodal displacement vector on the plane element divided on region A2 is denoted as δ i , the interpolation shape function within the plane element is denoted as N i , and the thickness stretching mode displacement function at any point within the plane element is expressed as:

[0036]

[0037] The energy functional of the plane element is expressed as:

[0038]

[0039] Among them,

[0040]

[0041] K i and M i are the stiffness matrix and mass matrix of the plane element .

[0042] In some embodiments, according to the principle of minimum potential energy, the energy of each plane element satisfies:

[0043]

[0044] Obtaining:

[0045]

[0046] Assembling all plane elements to obtain

[0047] [K - M(ω 2 )]δ = 0, (14)

[0048] Among them, K and M are the global stiffness matrix and global mass matrix respectively, and δ represents the global nodal displacement vector;

[0049] To solve for the non - zero nodal displacement vector, the determinant of the coefficient matrix in Equation (14) should be 0, that is

[0050] |K - M(ω 2 )| = 0 (15)

[0051] Equation (15) is the control equation for solving the FBAR resonance frequency ω. After determining the resonance frequency, substituting the resonance frequency into Equation (14) can determine the displacement vector of each node, thereby obtaining the displacement vibration mode f of the FBAR n 。

[0052] In some embodiments, in step S4, it specifically includes:

[0053] The coordinates of three nodes are P1(x1, y1), P2(x2, y2) and P3(x3, y3) respectively. The coordinates P(x, y) of any node within the triangular element are expressed in terms of the shape functions and the coordinates of the three nodes as:

[0054]

[0055] The three nodes P1, P2, P3 are sorted counter - clockwise, and the area of the triangular element is expressed as:

[0056]

[0057] Then the area coordinates corresponding to any point within the triangular element are expressed as

[0058]

[0059] The shape functions of any point within the triangular element are expressed in terms of the area coordinates, that is:

[0060]

[0061] According to the coordinates of the three nodes of the triangular element, the coefficient matrix of Equation (19) is solved; according to Equation (19), we get

[0062]

[0063] Therefore, we have

[0064]

[0065] Then according to Equations (9) and (10), we get:

[0066]

[0067]

[0068] Finally, the resonant frequency of the FBAR is determined according to formulas (15), (22), and (23). Further, according to formula (14), the displacement vectors of each node are solved to determine the modal vibration mode of the FBAR.

[0069] The beneficial effects of the present invention are as follows:

[0070] The present invention discloses a method for imaging the singular modal vibration mode of an FBAR. Based on the semi-analytical finite element theory, it is used to characterize the modal vibration mode of a three-dimensional FBAR at ultra-high frequencies. The three-dimensional modal vibration imaging of a three-dimensional FBAR device is reduced to two-dimensional planar imaging. This not only makes the original three-dimensional elements become simple planar elements, but also the number of elements in the overall structure simulation is reduced from the original order of 10 7 to the order of 10 4 The simplification of the element type and the significant reduction in the number of elements make the imaging technology provided by the present invention have a lower time cost. At the same time, it does not require various parameter settings and post-processing in commercial finite element software, realizing fast and quantitative imaging of the FBAR modal vibration mode. On the other hand, through triangular elements, any driving electrode configuration can be meshed, including rectangles, circles, triangles, and even irregular pentagons. Therefore, the method disclosed by the present invention can quickly and quantitatively image the singular modal vibration mode of a complex driving electrode type FBAR. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 is the front view of the FBAR of the present invention and the top view with different electrode configurations;

[0072] Figure 2 is a schematic diagram of the triangular element of the present invention;

[0073] Figure 3 is the modal vibration mode and frequency of the first-order thickness stretching mode of an FBAR with a rectangular driving electrode;

[0074] Figure 4 is the modal vibration mode and frequency of the first-order thickness stretching mode of an FBAR with a circular driving electrode

[0075] Figure 5 is the modal vibration mode and frequency of the first-order thickness stretching mode of an FBAR with a triangular driving electrode;

[0076] Figure 6 is the modal vibration mode and frequency of the first-order thickness stretching mode of an FBAR with an irregular pentagonal driving electrode. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0077] The following describes exemplary embodiments of the present disclosure with reference to the accompanying drawings. Various details of the embodiments of the present disclosure are included to facilitate understanding, and they should be considered merely exemplary. Therefore, those of ordinary skill in the art should recognize that various changes and modifications can be made to the embodiments described herein without departing from the scope and spirit of the present disclosure. Similarly, descriptions of well-known functions and structures are omitted in the following description for clarity and conciseness.

[0078] Please refer to Figure 1-2 , an imaging method for the singular mode vibration mode of FBAR based on the semi - analytical finite element theory is disclosed in an embodiment of the present invention, including the following steps:

[0079] Step S1, constructing a semi - analytical finite element formulation for FBAR with a complex electrode configuration;

[0080] Step S2, meshing the upper surface of the FBAR using triangular elements to obtain the coordinates of three nodes in each mesh element;

[0081] Step S3, determining the interpolation shape function, stiffness matrix, and mass matrix of each mesh element according to the node coordinates;

[0082] Step S4, assembling the total stiffness matrix and total mass matrix based on the stiffness matrix and mass matrix of each mesh element, and solving to obtain the resonant frequency of the FBAR;

[0083] Step S5, calculating the displacement vectors of all nodes in each mesh element according to the resonant frequency of the FBAR, and combining with the interpolation shape function of each mesh element to obtain the thickness - stretch mode displacement of each node of the FBAR, and drawing the mode vibration mode corresponding to the resonant frequency.

[0084] According to the report in the Ti ersten literature, the high - frequency vibration characteristics of the thickness - stretch working mode of FBAR can be characterized by a set of two - dimensional scalar equations;

[0085] For region A1 on the upper surface of the FBAR where the driving electrode is plated, the two - dimensional scalar equation is:

[0086]

[0087] For region A2 on the upper surface of the FBAR where the driving electrode is not plated, the two - dimensional scalar equation is

[0088]

[0089] Among them, M n represents the coefficient of the two - dimensional scalar equation corresponding to the n - th thickness - stretch mode. When n = 1, it represents the 1 - st thickness - stretch mode; f ndenotes the displacement component of the nth-order thickness-stretching mode along the z coordinate in the thickness direction, where x and y denote the coordinate components in two in-plane directions, and ρ f denotes the mass density of the piezoelectric thin film in the FBAR, ω denotes the resonance frequency of the nth-order thickness-stretching mode of the FBAR, and ω n denote the resonance frequency of the nth-order thickness-stretching mode of an infinitely large FBAR completely covered with the upper electrode and the resonance frequency of the nth-order thickness-stretching mode of an infinitely large FBAR completely uncovered with the upper electrode, respectively.

[0090] According to the above control equations corresponding to different upper electrode regions, with the upper drive electrode region being A1 and the region without the upper electrode being A2, the variational functional of the overall FBAR structure can be constructed and expressed as:

[0091]

[0092] If the cross-sections of region A1 and region A2 are divided into n1 and n2 elements respectively, the total energy of all elements is expressed as:

[0093]

[0094] where,

[0095]

[0096] the nodal displacement vector on the planar element divided on region A1 is denoted as the interpolation shape function within the planar element is denoted as the thickness-stretching mode displacement function at any point within the planar element is expressed as:

[0097]

[0098] the energy functional of the planar element is expressed as:

[0099]

[0100] where,

[0101]

[0102] and are the stiffness matrix and mass matrix of the planar element respectively.

[0103] the nodal displacement vector on the planar element divided on region A2 is denoted as δ i, the interpolation shape function in the plane element is expressed as N i , the thickness stretching mode displacement function at any point in the plane element is expressed as:

[0104]

[0105] Plane element The energy functional is expressed as:

[0106]

[0107] Among them,

[0108]

[0109] K i and M i are the stiffness matrix and mass matrix of the plane element .

[0110] According to the principle of minimum potential energy, the energy of each plane element satisfies:

[0111]

[0112] It is obtained that:

[0113]

[0114] Assembling all plane elements gives

[0115] [K - M(ω 2 )]δ = 0, (14)

[0116] Among them, K and M are the global stiffness matrix and global mass matrix respectively, and δ represents the global nodal displacement vector;

[0117] To solve for the non - zero nodal displacement vector, the determinant of the coefficient matrix of formula (14) should be 0, that is

[0118] |K - M(ω 2 )| = 0 (15)

[0119] Formula (15) is the control equation for solving the FBAR resonance frequency ω. After determining the resonance frequency, substituting the resonance frequency into equation (14), the displacement vector of each node can be determined, and thus the displacement vibration mode f n of the FBAR can be obtained.

[0120] For the FBAR structure, a general triangular element can be used to mesh the upper surface. The triangular element and node sorting are as Figure 2 shown:

[0121] The coordinates of the three nodes are P1(x1, y1), P2(x2, y2) and P3(x3, y3) respectively. The coordinates P(x, y) of any node within the triangular element are expressed in terms of the shape functions and the coordinates of the three nodes as follows:

[0122]

[0123] The three nodes P1, P2, P3 are sorted counterclockwise, and the area of the triangular element is expressed as:

[0124]

[0125] Then the area coordinates corresponding to any point within the triangular element are expressed as

[0126]

[0127] The shape functions of any point within the triangular element are expressed in terms of the area coordinates, i.e.:

[0128]

[0129] Based on the coordinates of the three nodes of the triangular element, the coefficient matrix of formula (19) is solved; according to formula (19), we get

[0130]

[0131] Therefore, we have

[0132]

[0133] Then according to formulas (9) and (10), we get:

[0134]

[0135]

[0136] Finally, according to formulas (15), (22) and (23), the resonant frequency of the FBAR is determined. Further, according to formula (14), the displacement vectors of each node are solved to determine the modal vibration mode of the FBAR.

[0137] The following are specific test examples of the present invention:

[0138] The structure of the FBAR is as Figure 1 shown, where the thickness of the piezoelectric thin film is 15 microns, the material is zinc oxide, the electrode material is a gold electrode, the mass ratio of the upper driving electrode to the piezoelectric thin film is 0.1, the mass ratio of the bottom electrode to the piezoelectric thin film is 0.0453, and the thickness of the silicon substrate is 5 microns;

[0139] Therefore, in the two-dimensional scalar equation, M1 = 3.17762×10 11 , and ω1 are equal to 1069.14839 MHz and 1119.032304 MHz respectively, where the thickness-stretch mode order n is taken as 1, that is, the first-order thickness-stretch mode.

[0140] In the present invention, the driving electrodes are respectively considered to be rectangular, circular, triangular, and irregular pentagonal in shape. The present invention performs mesh division on the corresponding FBAR through Python language software, so as to obtain the specific coordinate values of the three nodes of each mesh unit.

[0141] Determine the interpolation shape function of the mesh unit, as well as the stiffness matrix and mass matrix of each mesh unit according to formulas (18), (22), and (23).

[0142] Based on the obtained stiffness matrix and mass matrix of each mesh unit, assemble the total stiffness matrix and total mass matrix of the overall structure, and solve for the resonance frequency of the FBAR according to equation (15).

[0143] Substitute the solved resonance frequency into formula (14), solve for the displacement vectors of all corresponding units, and combine the obtained interpolation shape function to obtain the thickness-stretch mode displacement of each node of the FBAR, and then the mode shape corresponding to this resonance frequency can be plotted.

[0144] Through the numerical method disclosed in the present invention, the modes and frequencies of the FBAR with a rectangular driving electrode obtained by simulation analysis are as Figure 3 shown.

[0145] The modes and frequencies of the FBAR with a circular driving electrode obtained by simulation analysis are as Figure 4 shown.

[0146] The modes and frequencies of the FBAR with a triangular driving electrode obtained by simulation analysis are as Figure 5 shown.

[0147] The modes and frequencies of the FBAR with an irregular pentagonal driving electrode obtained by simulation analysis are as Figure 6 shown.

[0148] The present invention is based on the semi-analytical finite element theory to characterize the mode shape of the three-dimensional FBAR at ultra-high frequencies, and image the three-dimensional modal vibration of the three-dimensional FBAR device to two-dimensional plane imaging. This not only makes the original three-dimensional elements become simple plane elements, but also reduces the number of elements in the overall structure simulation from the original 10 7 order of magnitude to 10 4The order of magnitude, the simplification of the element type, and the significant reduction in the number of elements make the imaging technology provided by the present invention have a lower time cost. At the same time, it does not require various parameter settings and post-processing in commercial finite element software, achieving fast and quantitative imaging of the modal vibration modes of FBAR. On the other hand, any driving electrode configuration can be meshed through triangular elements, including rectangles, circles, triangles, and even irregular pentagons. Therefore, the method disclosed in the present invention can quickly and quantitatively image the singular modal vibration modes of complex driving electrode type FBARs.

[0149] The above specific embodiments do not constitute a limitation on the protection scope of the present disclosure. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principle of the present disclosure shall be included within the protection scope of the present disclosure.

Claims

1. An imaging method for FBAR singular mode shapes based on semi-analytical finite element theory, characterized in that: The following steps are involved: Step S1, constructing a semi-analytical finite element formula of a complex electrode configuration FBAR; Step S2, meshing the upper surface of the FBAR using triangular units, and obtaining the coordinates of three nodes in each mesh unit; Step S3, determining the interpolation shape function, stiffness matrix and mass matrix of each grid element according to the node coordinates; Step S4, assembling a total stiffness matrix and a total mass matrix based on the stiffness matrix and the mass matrix of each grid unit, and solving to obtain the resonant frequency of the FBAR; Step S5, according to the resonant frequency of the FBAR, calculate the displacement vectors of all nodes in each grid unit, combine the interpolation shape function of each grid unit, obtain the thickness tensile modal displacement of each node of the FBAR, and draw the modal vibration shape corresponding to the resonant frequency; In step S1, it specifically includes: The two-dimensional scalar equation for the area A1 on the upper surface of the FBAR where the driving electrode is plated is: The two-dimensional scalar equation for the area A2 on the FBAR top surface where the driving electrode is not plated is: Among them, M n represents the coefficient of the two-dimensional scalar equation corresponding to the n-th order thickness tensile mode, and when n = 1, it represents the 1-th order thickness tensile mode; f n represents the displacement component of the nth-order thickness tensile mode along the thickness direction coordinate component z, x and y represent the coordinate components in the two directions in the plane, respectively, ρ f represents the mass density of the piezoelectric film in the FBAR, ω represents the resonant frequency of the n-th order thickness stretch mode of the FBAR, and ω n They respectively represent the resonant frequency of the nth thickness stretching mode of the infinite FBAR completely covered with the upper electrode and the resonant frequency of the nth thickness stretching mode of the infinite FBAR completely uncovered with the upper electrode.

2. The imaging method of FBAR singular mode shapes based on semi-analytical finite element theory according to claim 1, characterized in that: In step S2, it also includes: The variational functional of the overall structure of FBAR is constructed as: The cross sections of area A1 and area A2 are divided into n1 and n2 units respectively, and the total energy of all units is expressed as: in, 3. The imaging method of FBAR singular mode vibration shape based on semi-analytical finite element theory according to claim 2, characterized in that: In step S2, it also includes: Plane units divided on area A1 The node displacement vector on The interpolation shape function of the plane element is expressed as Thickness tensile modal displacement function f at any point in the plane element n i It is expressed as: Planar unit A1 i The energy functional of is expressed as: in, and For planar units The stiffness matrix and mass matrix of .

4. The imaging method of FBAR singular mode vibration shape based on semi-analytical finite element theory according to claim 3, characterized in that: In step S2, it also includes: Plane units divided on area A2 The node displacement vector on is represented by δ i , the interpolation shape function in the plane element is expressed as N i , the thickness tensile modal displacement function at any point in the plane element is expressed as: Plane unit The energy functional of is expressed as: in, K i and M i For planar units The stiffness matrix and mass matrix of .

5. The imaging method of FBAR singular mode shapes based on semi-analytical finite element theory according to claim 4, characterized in that: According to the principle of minimum potential energy, the energy of each plane unit satisfies: get: Assemble all the planar units to get [KM(ω 2 )]δ=0, (14) Among them, K and M are the overall stiffness matrix and the overall mass matrix respectively, and δ represents the overall node displacement vector; To solve for non-zero node displacement vectors, the coefficient matrix determinant of formula (14) should be 0, that is, |KM(ω 2 )|=0 (15) Formula (15) is the control equation for solving the FBAR resonant frequency ω. After determining the resonant frequency, substitute the resonant frequency into equation (14) to determine the displacement vector of each node, thereby obtaining the displacement vibration mode f of the FBAR. n .

6. The imaging method of FBAR singular mode shapes based on semi-analytical finite element theory according to claim 5, characterized in that: In step S4, it specifically includes: The coordinates of the three nodes are P1(x1, y1), P2(x2, y2) and P3(x3, y3). The coordinates P(x, y) of any node in the triangle unit are expressed by the shape function and the coordinates of the three nodes as follows: The three nodes P1, P2, and P3 are arranged counterclockwise, and the area of ​​the triangular unit is expressed as: Then the area coordinates corresponding to any point in the triangular element are expressed as The shape function of any point in the triangular element is expressed by area coordinates, namely: According to the coordinates of the three nodes of the triangular unit, the coefficient matrix of formula (19) is obtained; according to formula (19), we get Therefore, there is Then according to formulas (9) and (10), we get: Finally, the resonant frequency of the FBAR is determined according to formulas (15), (22) and (23), and the displacement vector of each node is further solved according to formula (14), thereby determining the modal vibration shape of the FBAR.

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