A method for analyzing the propagation characteristics of a stacked surface acoustic wave

Abstract

This invention discloses a method for analyzing the propagation characteristics of multilayer surface acoustic waves (SAWs), comprising the following steps: obtaining material parameters of each layer and substrate material, including elastic coefficient, piezoelectric coefficient, dielectric constant, and density; setting structural parameters of each layer; performing dimensional analysis-based normalization on the parameters; calculating the coefficient matrix, transmission matrix, substrate mode matrix, and system matrix; applying boundary conditions to the top layer surface to obtain the system matrix after applying the boundary conditions and calculating the determinant value; performing a velocity scan and repeating the above steps to obtain the relationship curve between velocity and the determinant value of the boundary condition system matrix; finding the zero point of the relationship curve, the velocity corresponding to the zero point being the SAW phase velocity; calculating the electrical coupling coefficient; calculating state vector, mode shape, penetration depth, and other indicators; modifying structural parameters or temperature, and repeating the above steps to obtain the relationship between SAW propagation characteristics and the changes of each parameter.

Description

Technical Field

[0001] This invention relates to the field of surface acoustic wave (SAW) device theory, and specifically to a method for analyzing the propagation characteristics of stacked surface acoustic waves. Background Technology

[0002] Surface acoustic wave (SAW) devices are widely used in mobile communications, broadcast filtering, and sensor networks due to their advantages such as small size, low insertion loss, good selectivity, flexible design, and compatibility with semiconductor processes. With the development of 5G technology and the popularization of the Internet of Things (IoT), the market has placed higher demands on the operating frequency, bandwidth, and temperature stability of SAW devices.

[0003] To meet high-performance requirements, SAW devices often employ a multilayer thin-film structure deposited on a single-crystal substrate, including temperature compensation layers (SiO2), acoustic reflection layers, and trap layers. In this multilayer structure, the overall device performance is influenced by the combined advantages of each layer, and the corresponding SAW propagation characteristics (such as phase velocity, electromechanical coupling coefficient, and mode shape) are directly affected by the structural characteristics of each layer. Therefore, rapidly and accurately determining the propagation characteristics of multilayer SAW devices is a crucial aspect of device design.

[0004] Currently, the mainstream method for analyzing the propagation characteristics of various SAW device structures is the Finite Element Method (FEM). The FEM offers advantages such as high accuracy, strong adaptability, and the ability to handle complex geometries and boundary conditions. Furthermore, users do not need a deep understanding of piezoacoustics theory to operate it. However, FEM suffers from problems such as high computational cost, complex post-processing, and unclear physical images. Especially when dealing with multi-layered complex structures, modeling becomes complex and time-consuming, hindering the rapid understanding of the theoretical propagation characteristics of multi-layered structures and the determination of parameter trends.

[0005] The transmission matrix method, combined with partial wave theory, is a theoretical analysis approach proposed by piezoacoustics to address traditional SAW and multilayer SAW structures. The former transforms characteristic analysis into a matrix multiplication problem, while the latter treats the SAW as a superposition of several fundamental modes. If the extremely high computing power of today's computers can be leveraged to rapidly obtain the propagation characteristics of multilayer SAW structures through programming, it will greatly accelerate the theoretical exploration of device structures and achieve a rapid transformation from theory to application.

[0006] Existing technical literature derives a transfer matrix method applied to acoustic waves in multilayer structures, capable of handling piezoelectric layers and various electrical boundary conditions. However, it does not discuss numerical instability issues arising during calculation, and its system analysis capabilities for critical parameters in SAW device design, such as the electromechanical coupling coefficient (K²) and temperature stability, are limited. Yu TH Zhen's "Reflection and Transmission Analysis of Surface Acoustic Wave Devices" employs an eight-dimensional matrix method based on the Stroh form to analyze SAW propagation in piezoelectric multilayer media, using the transfer matrix method combined with the continuity condition of the grid edge to calculate the reflectivity and transmittance of the SAW. The study systematically analyzes the influence of geometric parameters such as grid width, grid height, and the number of grid bars on reflection / transmission characteristics, but it does not discuss the algorithm's programming implementation, the calculation of key parameters for SAW device design, or the temperature stability of the structure. Based on previous transmission matrix methods, this invention achieves SAW propagation characteristic analysis of arbitrary multi-layer structures through programming, parameterization, and normalization, and quickly determines the structural parameters of the optimal index through parameterized scanning. Summary of the Invention

[0007] To address the problems existing in the prior art, this invention proposes a method for analyzing the propagation characteristics of stacked surface acoustic waves (SAWs). This method, by constructing state-space equations and transmission matrices and combining them with partial wave theory, efficiently and accurately obtains the propagation characteristics and key performance indicators of SAWs.

[0008] The present invention is achieved by at least one of the following technical solutions.

[0009] A method for analyzing the propagation characteristics of stacked surface acoustic waves includes the following steps: S1. Obtain the material parameters of each layer material and the substrate material, including elastic modulus, piezoelectric coefficient, dielectric constant and density; S2. Set the structural parameters of each layer of material, including Euler angles and layer thickness based on ZXZ rotation, and perform coordinate rotation transformation on the material parameters; S3. Perform dimensional analysis-based normalization on the parameters to avoid singularities during subsequent matrix operations; S4. Calculate the coefficient matrix of each layer material and the substrate material; S5, based on coefficient matrix and layer thickness transmission matrix; S6. Solve for the eigenvalues ​​and eigenvectors of the coefficient matrix of the substrate material, and select the eigenvectors that meet the requirements according to the radiation conditions to form the substrate mode matrix. Construct the substrate surface state vector through the substrate mode matrix. S7. Calculate the system matrix based on the substrate mode matrix; S8. Apply boundary conditions to the top surface, including mechanical boundary conditions with zero normal stress and electrical boundary conditions with open or short circuits, to obtain the system matrix after applying the boundary conditions and calculate the determinant value. S9. Perform a velocity scan and repeat steps S3 to S8 to obtain the relationship curve between velocity and the determinant value of the boundary condition system matrix. S10. Find the zero point of the relationship curve according to the zero point search method. The velocity corresponding to the zero point is the SAW phase velocity. S11, Computer electrical coupling coefficient; S12. Calculate the state vector, mode shape, and penetration depth index; S13. Modify the corresponding parameters and repeat steps S3 to S12 to perform parameter analysis and obtain the relationship between the SAW propagation characteristics and the changes of each parameter.

[0010] Furthermore, in step S3, the normalization process uses five physical parameters—velocity, frequency, density, dielectric constant, and temperature—as normalization reference quantities. The normalization reference values ​​of other physical quantities are all derived from the reference values ​​and dimensional relationships of the above five physical parameter normalization reference quantities.

[0011] Furthermore, in step S4, the coefficient matrix is ​​the coefficient matrix of the state-space equation with respect to the state vector ξ, where the state vector is defined as ξ = [U, Φ, T, D]. , where U is the displacement component, Φ is the electric potential, T is the stress component, and D is the electric displacement component.

[0012] Further, in step S6, the radiation condition screening rule is as follows: eigenvalues ​​with a real part less than 0 are attenuated waves and are retained; eigenvalues ​​with a real part equal to 0 are propagating waves, and based on the direction of complex power energy flow, waves with a negative real part of the energy flow are retained.

[0013] Furthermore, the formula for calculating the system matrix U in step S7 is: U = P v , P is the total transmission matrix obtained by multiplying the transmission matrices of each layer. v The substrate mode matrix; Furthermore, the substrate surface state vector is =P v C v C v This represents the vector of undetermined coefficients.

[0014] Furthermore, the zero-point search method in step S10 is the second-order difference search method, that is, first find the minimum point, and then select the points whose second-order difference on both sides is negative and whose absolute value exceeds the threshold as zero points.

[0015] Furthermore, in step S13, the parameters of frequency, Euler angle, layer thickness, and temperature are modified, and steps S4 to S12 are repeated to obtain the relationship between the SAW propagation characteristics and the frequency and structural parameters.

[0016] Furthermore, in step S13, the temperature coefficient TC of the material and the thermal expansion coefficient α of the substrate are additionally obtained in step S1. Steps S3 to S11 are repeated at different temperatures, and the time-delay temperature coefficient is calculated. :

[0017] in It is the phase velocity, and T is the temperature.

[0018] Further, in step S13, the nontrivial solution of the undetermined coefficient vector of the substrate surface state vector is calculated by SVD, the displacement field distribution is obtained by the undetermined coefficient vector, and the root mean square processing of the displacement field is performed to calculate the penetration depth.

[0019] Compared with existing technologies, the beneficial effects of the present invention are as follows: This invention combines the transmission matrix method and partial wave theory to transform the complex analysis problem of multilayer SAW structures into matrix operations, offering advantages such as high computational efficiency, strong versatility, and clear physical concepts. This method is applicable to stacked structures of any number of layers and any materials, providing rapid and accurate theoretical guidance for SAW device design and possessing high practical value. Attached Figure Description

[0020] Figure 1 This is a schematic flowchart illustrating a method for analyzing the propagation characteristics of stacked surface acoustic waves, as described in an embodiment. Figure 2 This is a schematic diagram illustrating the structure and coordinate definition of a multilayer SAW device in an embodiment. Figure 3 This example illustrates the logarithmic relationship curve of the speed-system matrix determinant and a diagram of zero-point search. Figure 4a This is a schematic diagram of the speed-frequency relationship curve for an example. Figure 4b This is a schematic diagram of the speed-frequency relationship curve for an example. Figure 4c This is a schematic diagram of the speed-frequency relationship curve for an example. Figure 5 This is a schematic diagram of an Euler angle scan for an embodiment; Figure 6a This is a schematic diagram of the thickness scan for mode 1 in the embodiment; Figure 6b This is a schematic diagram of the thickness scan for mode 2 in the embodiment; Figure 6c This is a schematic diagram of the thickness scan for mode 3 in the embodiment; Figure 7 This is a schematic diagram of temperature scanning for an example.

[0021] Figure 8 The relationship between phase velocity and temperature for the LN / Si structure is presented. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will now be described in further detail with reference to the accompanying drawings.

[0023] like Figure 1 As shown, the method for analyzing the propagation characteristics of stacked surface acoustic waves in this embodiment is performed in Matlab and includes the following steps: Step S1: Obtain and convert material parameters.

[0024] Obtain the material parameters of each layer and the substrate material, including the elastic coefficient c, piezoelectric coefficient e, dielectric constant ε, and density ρ.

[0025] like Figure 2 The layer diagram shown defines the labeling method for the substrate and each thin film layer: Any N thin films (Layer 1, Layer 2, Layer 3, ..., Layer N) are stacked on the substrate (Sub), with layer thicknesses h1, h2, h3, ..., h... N The axial direction and origin of the depth coordinate axis are also defined: the axial direction is the surface normal of the layer, pointing from the substrate to the thin film; the origin is set at the substrate surface.

[0026] The material parameters refer to the three main parameters c, e, and ε, specifically in three forms: simplified array form, Voigt matrix form, and complete tensor form. Depending on the computational requirements, these three forms need to be converted to achieve the highest computational efficiency. For example, temperature coefficient calculation is simplest in the simplified array form, rotation transformation is more efficient in the Voigt form, and the construction of the coefficient matrix requires the tensor form.

[0027] Preferably, the conversion of parameter forms is mediated by the Voigt matrix form.

[0028] Step S2: Setting structural parameters and coordinate rotation transformation.

[0029] The structural parameters of each layer of material include layer thickness. Euler angles (α, β, γ) of ZXZ rotation. The rotation rule for Euler angles is: the original coordinate axis is first rotated around the original Z-axis to an angle α, then around the new X-axis to an angle β, and finally around the new Z-axis to an angle γ, thus obtaining the new coordinate axis.

[0030] The rotational transformation of the material parameters is performed in Voigt form, using the rotation matrix A and the Bond matrix. This involves transforming three main parameters. The rotation matrix A refers to the coordinates of a point on the original coordinate axes. Coordinates transformed to the new coordinate axis Transformation matrix: Bond matrix The coefficients are obtained by transforming the elements of the rotation matrix A using the elastic coefficient c and the piezoelectric coefficient e. , , :

[0031] Step S3: Parameter normalization.

[0032] The material and structural parameters are normalized to avoid instability in the calculation of ill-conditioned or exponential matrices caused by the difference in magnitude between parameters such as elastic coefficient (GPa level) and layer thickness (hundreds of nanometers level).

[0033] According to dimensional analysis theory, physical laws are scale-independent, and linear scaling of a single dimension does not change the physical equations. The basic dimensions of the physical quantities involved in this invention include length L, time T, mass M, current I, and temperature Θ.

[0034] According to Buckingham's π theorem, five independent normalized reference quantities need to be selected. As an example, these are velocity (L / T²), frequency (1 / T), density (M / L³), and dielectric constant (I²T). 4 L (L / ML³) and T (T), M, I, and Θ represent the dimensions of length, time, mass, current, and temperature, respectively.

[0035] The normalized reference values ​​for all other physical quantities (including length, period, angular frequency, elastic coefficient, piezoelectric coefficient, wavenumber, etc.) are calculated from the reference values ​​and dimensional relationships of the above five reference quantities. It is important to note that the dimensions of angular frequency ω and frequency f are both 1 / T; therefore, their normalized reference values ​​should be equal, and the relationship f = 2πω cannot be used for conversion.

[0036] The normalized value of a parameter is obtained by dividing the actual value of all parameters by the normalized reference value of the corresponding physical quantity, which is then used for subsequent matrix operations.

[0037] Step S4: Construct the coefficient matrix M.

[0038] Based on piezoacoustic theory and the plane wave hypothesis, a coefficient matrix M is constructed for each layer of material and the substrate material. This matrix is ​​the coefficient matrix of the state-space equation (1), and its specific expression is obtained by matrix operations on the elastic coefficient, piezoelectric coefficient, dielectric constant, density, angular frequency ω, and wave number κ of the material.

[0039] The coefficient matrix M is the coefficient matrix of the state-space equations with respect to the state vector ξ: (1) Wherein, the state vector The definition of is: (2) In the formula , , All are displacement components, with subscripts 1, 2, and 3 indicating... , , Three directions, Corresponding to the SAW propagation direction, Corresponding to the surface normal of the multi-layer structure, and , Construct a right-handed rectangular coordinate system; It is the electric potential; , , All are stress components. Depth coordinates The electric displacement component. The coordinate axes are defined as follows: Figure 2 .

[0040] The general solution of equation (1) is: (1-1) in and These represent the state vector in depth coordinates. and The value, That is, the depth difference; Equation (1-1) shows that the state vector of any depth difference can be obtained by left-multiplying by a factor equal to the depth difference. Related matrix Establish contact, This represents the coefficient matrix.

[0041] The above formula is also the direct source of the transfer matrix calculation formula (3).

[0042] Step S5: Calculate the transmission matrix for each layer.

[0043] For thel Layer transfer matrix It can be obtained through the corresponding layer coefficient matrix and layer thickness calculate: (3) in l The number of layers in the thin film. l The values ​​can be 1, 2, 3, ..., N.

[0044] The total transmission matrix is ​​obtained by multiplying the transmission matrices of each layer. : (4) in N It is the total number of floors. Total transmission matrix The state vector connected to the substrate surface and the state vector of the top surface : .

[0045] in and Representing depth coordinates and The state vector, where H and 0 are the depth coordinates of the substrate surface of the top layer, respectively. .

[0046] The transmission matrix of each layer is obtained through equation (3). The total transmission matrix is ​​obtained through equation (4). .

[0047] Step S6: Construct the substrate mode matrix: Based on partial wave theory, solve for the coefficient matrix of the substrate material (referred to as the substrate coefficient matrix). The eigenvalues ​​and eigenvectors are determined, and eigenvectors that meet the requirements are selected based on the radiation conditions to form the substrate mode matrix. ; Based on partial wave theory, the state vector of the substrate can be represented as the substrate coefficient matrix. Linear superposition of eigenvectors: (1-2) In the above formula, the subscript j The first element of the substrate coefficient matrix represents the... j Each eigenvalue-eigenvector pair and for The eigenvalues ​​and eigenvectors of a given molecule. (This is called...) It is a partial wave. C j It is the firstj The coefficients of each partial wave.

[0048] Depending on the radiation conditions, part of the SAW wave will travel along - x 3-directional attenuation or propagation: Attenuation: If real part If the value is greater than 0, then part of the wave is an attenuated wave, directly satisfying the radiation condition; Dissemination: If If the real part is 0, then the partial wave is a propagating wave. It can only flow along... Negative propagation waves satisfy the radiation condition.

[0049] According to piezoacoustic theory Directional energy flow The calculation formula is: (1-3) in, yes Stress in the direction, It is a displacement field. It is the imaginary unit, ω=2πf is the angular frequency, and the superscript is... This indicates taking the complex conjugate of the matrix.

[0050] There are exactly four partial waves that satisfy the radiation conditions, and these eigenvectors constitute the substrate mode matrix. (8×4 matrix), the eigenvalues ​​corresponding to these eigenvectors constitute the eigenvalue matrix Λ v The substrate surface state vector can be represented as: , v This indicates that the radiation conditions are met. It is a vector of coefficients to be determined, given by C. j The components that meet the radiation conditions.

[0051] Step S7: Define the system matrix U: Top surface state vector It can be represented as:

[0052] Step S8: Apply boundary conditions to the top surface. The mechanical boundary conditions require zero normal stress. (5) Represents the top surface value.

[0053] Electric boundary conditions are divided into two categories: Open circuit (continuous potential): (6) Short circuit (zero potential): (7) and These represent the state vectors of the top surface, respectively. and The value of ε0; where ε0 is the vacuum permittivity. κ Let be the wave number. After applying boundary conditions, the system matrix is ​​obtained. and ; Based on equations (5) and (6), establish open-circuit boundary conditions, and based on equations (5) and (7), establish short-circuit boundary conditions.

[0054] Preferably, the system matrix U is divided into row vectors: Then the boundary conditions can be expressed as: open circuit:

[0055] Short circuit:

[0056] in It is the vacuum permittivity. It is the wave number. When the determinant of the system matrix under the applied boundary conditions is 0, equation (8) has a nontrivial solution, and SAW exists.

[0057] (8) This represents the system matrix under open-circuit / short-circuit boundary conditions.

[0058] Step S9, Speed ​​Scan.

[0059] speed Perform a scan, and for each velocity point, repeat steps S3 to S8 to calculate the corresponding system matrix determinant and take its logarithm. This yields the logarithmic relationship curve between velocity and determinant.

[0060] Optionally, when the number of attenuated waves calculated in step 6 is less than 4, it means that a propagating wave has appeared, and at this point, volume leakage has occurred in the SAW. The velocity point at this time is the critical velocity for volume leakage to occur. .

[0061] Step S10, Zero Point Search Find the zero point of the logarithmic velocity-determinant relationship curve. Due to scanning accuracy limitations, not all minimum points of the logarithmic relationship curve are zero points of the determinant. In one embodiment, the second-order difference method is used for determination: when the extreme point is convex to the left and right (the second-order difference is negative) and the convexity exceeds a set threshold, it is determined to be a zero point. The velocity at this point is the phase velocity of the surface acoustic wave, denoted as . .

[0062] Step S11, Computer electrical coupling coefficient.

[0063] According to equation (9), the computer electrical coupling coefficient K 2 : (9) in These were obtained under open-circuit boundary conditions and short-circuit boundary conditions, respectively. .

[0064] Figure 3 This figure shows the logarithmic velocity-determinant relationship curves for the LT (lithium tantalate) / SiO2 (silicon dioxide) / Si (silicon) structure. Blue and red lines, upper and lower triangles, and O1~O3 and S1~S3 are used to mark the logarithmic relationship curves and SAW modes under open-circuit and short-circuit boundary conditions, respectively. The critical velocity points are marked with green dashed lines. The K values ​​for each mode are recorded in text boxes on the right side of the figure. 2 and phase velocity.

[0065] Step S12, Pattern Analysis.

[0066] Solve for C in equation (8) v Non-trivial solutions. For Perform Singular Value Decomposition (SVD):

[0067] Among them U svd and V svd It is for U open / short The unitary matrix obtained by performing singular value decomposition, Σ, is a subset of U. open / short The singular value diagonal matrix obtained by performing singular value decomposition then becomes the vector of undetermined coefficients. For V svd This corresponds to a linear combination of column vectors with singular values ​​of 0. We obtain... Afterwards, the state vector of the substrate can be obtained through... The state vectors of the stacked layers can be calculated using the transfer matrix and recursion from equation (1-1), thus obtaining the state vector distribution along the entire depth direction. Displacement field mode shapes can also be extracted.

[0068] The displacement fields in the three directions are processed using root mean square (RMS) methods:

[0069] in displacement field RMS value Calculate throughout the depth direction If it follows Negative attenuation allows for the definition of penetration depth. for The depth relative to the top surface when it decays to 1 / e of the top surface value is used to extract physical quantities such as penetration depth.

[0070] Figures 4a-4c yes Figure 3 Displacement field distribution of the three modes and Distribution, respectively corresponding Figure 3 Modes 1-3 are marked with blue, red, green, and black lines respectively. , , , With depth coordinates The curves showing the changes were displayed, and the penetration depth of each mode was marked with a pink dashed line. The value and in depth coordinates The location.

[0071] Step S13, Parameter Analysis Modify parameters such as frequency, Euler angles, layer thickness, and temperature, and repeat steps S4 to S12 to obtain the SAW propagation characteristics (phase velocity). electromechanical coupling coefficient The relationship between the temperature delay coefficient (TCD) and frequency and structural parameters.

[0072] Step S13 includes frequency scanning. Different frequencies can be input and steps S1 to S10 can be repeated to obtain the phase velocity. The relationship with frequency f is the dispersion characteristic of the device.

[0073] Figure 5 This shows the relationship between the SAW mode characteristics of the LN (lithium niobate) / Si structure and frequency. The solid and dashed lines correspond to the open-circuit and short-circuit modes, respectively, and the blue circles, red squares, yellow upper triangles, and purple lower triangles correspond to mode 1, mode 2, mode 3, and mode 4, respectively.

[0074] Step S13 includes structural parameter scanning. The Euler angles and layer thicknesses in S2 can be scanned. By setting different Euler angles or layer thicknesses and repeating steps S3 to S11, the relationship between each index and the structural parameters can be obtained.

[0075] Figures 6a-6c It is the SAW mode of the LT / SiO2 / Si structure. A three-dimensional scatter plot showing the variation of Euler angles (α,β,γ). Figure 6a , Figure 6b , Figure 6c Corresponding to Figure 3 Modes 1, 2, and 3 are represented by scatter plot colors and color bars to indicate the Euler angles corresponding to each coordinate. The highest value is highlighted with a five-pointed star. The location.

[0076] Figure 7 It is the SAW mode of the LN / Si structure. The relationship between LN thickness and mode 1 and mode 2 is shown in blue and red solid lines, respectively.

[0077] Step S13 includes a temperature scan. In step S1, the temperature coefficient TC of the material and the thermal expansion coefficient α of the substrate are additionally obtained. Steps S3 to S11 are repeated at different temperatures, and the time-delay temperature coefficient under each boundary condition is calculated using the following formula. : (10) Where T is temperature. It is the phase velocity under each boundary condition.

[0078] Figure 8 The phase velocity of the LN / Si structure as a function of temperature is presented. In the figure, the solid circles and dashed squares correspond to open-circuit and short-circuit boundary conditions, and the blue and red lines correspond to mode 1 and mode 2, respectively. The calculated results of the time delay temperature coefficient for each mode under each boundary condition are also given.

[0079] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. A method for analyzing the propagation characteristics of stacked surface acoustic waves, characterized in that, Includes the following steps: S1. Obtain the material parameters of each layer material and the substrate material, including elastic modulus, piezoelectric coefficient, dielectric constant and density; S2. Set the structural parameters of each layer of material, including Euler angles and layer thickness based on ZXZ rotation, and perform coordinate rotation transformation on the material parameters; S3. Perform dimensional analysis-based normalization on the parameters to avoid singularities during subsequent matrix operations; S4. Calculate the coefficient matrix of each layer material and the substrate material; S5, based on coefficient matrix and layer thickness transmission matrix; S6. Solve for the eigenvalues ​​and eigenvectors of the coefficient matrix of the substrate material, and select the eigenvectors that meet the requirements according to the radiation conditions to form the substrate mode matrix. Construct the substrate surface state vector through the substrate mode matrix. S7. Calculate the system matrix based on the substrate mode matrix; S8. Apply boundary conditions to the top surface, including mechanical boundary conditions with zero normal stress and electrical boundary conditions with open or short circuits, to obtain the system matrix after applying the boundary conditions and calculate the determinant value. S9. Perform a velocity scan and repeat steps S3 to S8 to obtain the relationship curve between velocity and the determinant value of the boundary condition system matrix. S10. Find the zero point of the relationship curve according to the zero point search method. The velocity corresponding to the zero point is the SAW phase velocity. S11, Computer electrical coupling coefficient; S12. Calculate the state vector, mode shape, and penetration depth index; S13. Modify the corresponding parameters and repeat steps S3 to S12 to perform parameter analysis and obtain the relationship between the SAW propagation characteristics and the changes of each parameter.

2. The method for analyzing the propagation characteristics of stacked surface acoustic waves according to claim 1, characterized in that, In step S3, the normalization process uses five physical parameters—velocity, frequency, density, dielectric constant, and temperature—as normalization reference values. The normalization reference values ​​of other physical quantities are obtained from the reference values ​​and dimensional relationships of the above five physical parameters.

3. The method for analyzing the propagation characteristics of stacked surface acoustic waves according to claim 1, characterized in that, In step S4, the coefficient matrix is ​​the coefficient matrix of the state-space equation with respect to the state vector ξ, which is defined as ξ = [U, Φ, T, D]. , where U is the displacement component, Φ is the electric potential, T is the stress component, and D is the electric displacement component.

4. The method for analyzing the propagation characteristics of stacked surface acoustic waves according to claim 1, characterized in that, Step S6, the radiation condition screening rule is: eigenvalues ​​with a real part less than 0 are attenuated waves and are retained; eigenvalues ​​with a real part equal to 0 are propagating waves, and based on the direction of complex power energy flow, waves with a negative real part of energy flow are retained.

5. The method for analyzing the propagation characteristics of stacked surface acoustic waves according to claim 1, characterized in that, The formula for calculating the system matrix U in step S7 is: U = P v , P is the total transmission matrix obtained by multiplying the transmission matrices of each layer. v is the substrate mode matrix.

6. The method for analyzing the propagation characteristics of stacked surface acoustic waves according to claim 5, characterized in that, The substrate surface state vector is =P v C v C v This represents the vector of undetermined coefficients.

7. The method for analyzing the propagation characteristics of stacked surface acoustic waves according to claim 1, characterized in that, The zero-point search method in step S10 is the second-order difference search method, that is, first find the minimum point, and then select the points whose second-order difference on both sides is negative and whose absolute value exceeds the threshold as the zero point.

8. The method for analyzing the propagation characteristics of stacked surface acoustic waves according to claim 1, characterized in that, In step S13, the parameters of frequency, Euler angle, layer thickness, and temperature are modified, and steps S4 to S12 are repeated to obtain the relationship between the SAW propagation characteristics and the frequency and structural parameters.

9. The method for analyzing the propagation characteristics of stacked surface acoustic waves according to claim 1, characterized in that, In step S13, the temperature coefficient TC of the material and the thermal expansion coefficient α of the substrate are additionally obtained in step S1. Steps S3 to S11 are repeated at different temperatures, and the time delay temperature coefficient is calculated. : in It is the phase velocity, and T is the temperature.

10. A method for analyzing the propagation characteristics of stacked surface acoustic waves according to any one of claims 1 to 9, characterized in that, In step S13, the nontrivial solution of the undetermined coefficient vector of the substrate surface state vector is calculated by SVD, the displacement field distribution is obtained by the undetermined coefficient vector, and the root mean square processing of the displacement field is performed to calculate the penetration depth.

Images