Method for analyzing thermal-mechanical-electric coupling of three-dimensional bulk acoustic wave resonator device based on fdti algorithm

Abstract

The application discloses a three-dimensional bulk acoustic wave resonator device thermal-force-electric coupling analysis method based on an FDTD algorithm, relates to the field of bulk acoustic wave resonators, and comprises the following steps: a three-dimensional film bulk acoustic wave resonator geometric model is established; parameter and boundary condition setting and hexahedral mesh division are performed on the three-dimensional film bulk acoustic wave resonator geometric model to obtain a cell structure; and a thermal-force-electric coupling equation is established based on the cell structure; wherein the thermal-force-electric coupling equation comprises a piezoelectric constitutive equation, an electric field equation, a mechanical motion equation, a heat conduction equation and a thermal expansion equation; and the thermal-force-electric coupling equation is solved by using a time domain difference finite method to obtain various physical parameters in the thermal-force-electric coupling equation. The application improves the calculation efficiency and calculation precision and is more in line with the actual characteristics of the film bulk acoustic wave resonator.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of thin film bulk acoustic resonator, and particularly relates to a three-dimensional bulk acoustic resonator device thermal-mechanical-electric coupling analysis method based on FDTD algorithm. BACKGROUND

[0002] A thin film bulk acoustic resonator (FBAR) is a core unit of a filter, and a core structure thereof is sandwiched between upper and lower electrodes by piezoelectric material (such as aluminum nitride AlN or zinc oxide ZnO), and high-frequency signal selection and filtering functions are achieved by exciting bulk acoustic wave resonance in the thickness direction. Compared with a traditional quartz crystal resonator, the FBAR has the advantages of small size, high Q value, large power capacity, and is particularly suitable for high-frequency and high-integration scenarios such as 5G communication and Internet of Things (IoT).

[0003] The resonant frequency (such as center frequency, temperature drift) and quality factor (Q value) of the FBAR and other key electrical performance parameters are easily affected by the external environment temperature. In the first aspect, the piezoelectric coefficient (d33), dielectric constant (ε), elastic constant (c) and the like of the piezoelectric material will change with temperature. For example, the piezoelectric coefficient of AlN decreases nonlinearly with the increase of temperature, resulting in a decrease in electromechanical coupling efficiency; the resistivity of the electrode material (such as molybdenum, aluminum) increases with the increase of temperature, which aggravates the ohmic loss and thus reduces the Q value. In the second aspect, the piezoelectric layer, electrode layer and substrate material (such as silicon, sapphire) constituting the FBAR have different coefficients of thermal expansion (CTE). When the temperature changes, the thermal mismatch between the layers will cause thermal stress in the device, resulting in structural bending or delamination, and thus changing the effective thickness of the resonator and the sound wave propagation path, causing the resonant frequency to shift and the insertion loss to deteriorate. In the third aspect, the increase of temperature will reduce the sound speed in the piezoelectric material (such as the temperature coefficient of sound speed of AlN is about -25 ppm / ℃), which directly affects the temperature coefficient of resonant frequency (TCF). At the same time, the increase of temperature will aggravate the lattice scattering effect in the sound wave propagation process, resulting in an increase in acoustic energy loss and further reducing the Q value.

[0004] The Chinese invention patent with the publication number CN117634418B discloses a three-dimensional bulk acoustic wave resonator circuit electric-force-thermal coupling characteristic analysis method, comprising: establishing a geometric model of a three-dimensional thin film bulk acoustic wave resonator, and performing parameter setting on the geometric model, setting boundary conditions of the geometric model and performing hexahedral mesh partitioning to obtain a divided cell structure; constructing an electric-force-thermal coupling equation, the thermal-force-electric coupling equation including a piezoelectric constitutive equation, an electrostatic field equation, a mechanical motion equation and a heat conduction equation, the mechanical motion equation being obtained by updating the original mechanical motion equation considering damping terms; solving the thermal-force-electric coupling equation by using a FDTD algorithm to obtain each physical parameter in the thermal-force-electric coupling equation. The invention considers the mechanical damping of piezoelectric materials, refines the mechanical motion equation and the piezoelectric constitutive equation, establishes a three-dimensional grid, and realizes efficient analysis of the electric-force-thermal coupling characteristics of the three-dimensional bulk acoustic wave resonator by combining the FDTD algorithm. However, the heat conduction equation in the patent does not consider the influence of the external environment temperature on the electrical properties of the FBAR. SUMMARY

[0005] To solve the technical problems in the background art, the present application proposes a three-dimensional bulk acoustic wave resonator device thermal-force-electric coupling analysis method based on the FDTD algorithm.

[0006] The three-dimensional bulk acoustic wave resonator device thermal-force-electric coupling analysis method based on the FDTD algorithm proposed by the present application comprises:

[0007] establishing a three-dimensional thin film bulk acoustic wave resonator geometric model;

[0008] performing parameter setting, boundary condition setting and hexahedral mesh partitioning on the three-dimensional thin film bulk acoustic wave resonator geometric model to obtain a cell structure;

[0009] based on the cell structure, establishing a thermal-force-electric coupling equation; wherein the thermal-force-electric coupling equation includes a piezoelectric constitutive equation, an electric field equation, a mechanical motion equation, a heat conduction equation and a thermal expansion equation;

[0010] solving the thermal-force-electric coupling equation by using a time domain finite difference method to obtain each physical parameter in the thermal-force-electric coupling equation.

[0011] Preferably, the heat conduction equation should be:

[0012] ;

[0013] wherein, denotes density, denotes constant pressure heat capacity, denotes thermal conductivity, denotes time, denotes divergence, denotes temperature, denotes Hamiltonian operator;

[0014] wherein, ; in which x, y and z are three parameters of a spatial rectangular coordinate system.

[0015] Preferably, the thermal expansion equation is:

[0016] ;

[0017] in which, denotes thermal strain, denotes temperature, denotes time, denotes thermal expansion coefficient at time and temperature , denotes reference temperature.

[0018] Preferably, the piezoelectric constitutive equation is: ;

[0019] in which, denotes stress first order tensor, denotes elastic strain, denotes electric displacement vector, denotes electric field intensity, denotes elastic constant second order tensor, denotes dielectric constant second order tensor, denotes piezoelectric stress constant second order tensor, denotes transpose of .

[0020] Preferably, the electric field equation is:

[0021] ;

[0022] in which, denotes divergence, denotes electric displacement vector, denotes electric field intensity, denotes electric potential, denotes Hamiltonian operator.

[0023] Preferably, the mechanical motion equation is:

[0024] ;

[0025] ;

[0026] ;

[0027] ;

[0028] wherein, represents a displacement vector, represents a density, represents a strain first-order tensor, and is an operator, represents a stress first-order tensor, represents time.

[0029] Preferably, the three-dimensional thin film bulk acoustic resonator geometric model comprises, from top to bottom, a first aluminum nitride protective layer, a first molybdenum electrode layer, a second aluminum nitride piezoelectric layer, a second molybdenum electrode layer, and a silicon dioxide substrate layer.

[0030] Preferably, the second aluminum nitride piezoelectric layer needs to satisfy a constitutive equation, an electric field equation, a mechanical motion equation, a heat conduction equation, and a thermal expansion equation;

[0031] The first molybdenum electrode layer and the second molybdenum electrode layer need to satisfy an electric field equation, a mechanical motion equation, a heat conduction equation, and a thermal expansion equation;

[0032] The first aluminum nitride protective layer and the silicon dioxide substrate layer need to satisfy a mechanical motion equation, a heat conduction equation, and a thermal expansion equation.

[0033] Preferably, the heat-force-electric coupling equation is solved by using a time-domain finite difference method to obtain various physical parameters in the heat-force-electric coupling equation, including:

[0034] The time-domain finite difference algorithm is used to represent the differential of the spatial and time variables in a difference form, and the calculation is directly performed through the difference equation;

[0035] When the time step iteration starts, the displacement vector is obtained from the mechanical motion equation , and the temperature change is obtained from the heat conduction equation to obtain the generated thermal strain ; according to the strain first-order tensor and the thermal strain , the elastic strain is obtained; the elastic strain is substituted into the Poisson equation in the electric field equation to solve the electric potential ; the electric potential and the elastic strain are brought into the piezoelectric constitutive equation to obtain the stress first-order tensor , and the stress first-order tensor is brought into the mechanical motion equation to solve the displacement vector of the next time, and then the iteration calculation is performed along with the time until the preset time is reached.

[0036] In the present application, the three-dimensional bulk acoustic resonator device thermal-force-electric coupling analysis method based on the FDTD algorithm is proposed, by considering the influence of external temperature on the electrical performance, a new thermal-force-electric coupling equation is constructed, wherein the thermal-force-electric coupling equation includes the piezoelectric constitutive equation, the electric field equation, the mechanical motion equation, the heat conduction equation and the thermal expansion equation, which provides a method for the working performance simulation of the bulk acoustic resonator at different temperatures, and the time domain difference finite method is used to solve the thermal-force-electric coupling equation, the physical parameters in the thermal-force-electric coupling equation are obtained, the calculation efficiency and the calculation accuracy are improved, and the actual characteristics of the film bulk acoustic resonator are more in line with the actual characteristics. BRIEF DESCRIPTION OF DRAWINGS

[0037] Figure 1 The structure diagram of the three-dimensional bulk acoustic resonator device thermal-force-electric coupling analysis method based on the FDTD algorithm in an embodiment of the present application is proposed.

[0038] Figure 2 The resonator current-time curve at normal temperature in an embodiment of the present application is proposed.

[0039] Figure 3 The resonator current-time curve at different temperatures in an embodiment of the present application is proposed. DETAILED DESCRIPTION

[0040] It should be noted that the embodiments in the present application and the features in the embodiments can be combined with each other without conflict. The present application will be described in detail below with reference to the drawings and in combination with the embodiments.

[0041] REFERENCE Figure 1 The three-dimensional bulk acoustic resonator device thermal-force-electric coupling analysis method based on the FDTD algorithm proposed in the present application comprises:

[0042] A three-dimensional film bulk acoustic resonator geometric model is established;

[0043] The three-dimensional film bulk acoustic resonator geometric model is subjected to parameter setting, boundary condition setting and hexahedral mesh partitioning to obtain a cell structure;

[0044] Based on the cell structure, a thermal-force-electric coupling equation is established; wherein the thermal-force-electric coupling equation includes the piezoelectric constitutive equation, the electric field equation, the mechanical motion equation, the heat conduction equation and the thermal expansion equation;

[0045] The time domain difference finite method is used to solve the thermal-force-electric coupling equation to obtain the physical parameters in the thermal-force-electric coupling equation.

[0046] This invention constructs a new thermo-mechanical-electric coupling equation by taking into account the influence of external temperature on electrical performance. The thermo-mechanical-electric coupling equation includes the piezoelectric constitutive equation, electric field equation, mechanical motion equation, heat conduction equation, and thermal expansion equation. It provides a method for simulating the working performance of bulk acoustic wave resonators at different temperatures. The thermo-mechanical-electric coupling equation is solved by the finite difference time-domain method to obtain the various physical parameters in the thermo-mechanical-electric coupling equation, which improves the calculation efficiency and accuracy and is more consistent with the actual characteristics of thin-film bulk acoustic wave resonators.

[0047] The geometric model of the three-dimensional thin-film bulk acoustic resonator in this embodiment includes, from top to bottom, a first aluminum nitride (AlN) protective layer, a first molybdenum (Mo) electrode layer, a second aluminum nitride (AlN) piezoelectric layer, a second molybdenum (Mo) electrode layer, and a silicon dioxide (SiO2) substrate layer.

[0048] Specifically, the second aluminum nitride piezoelectric layer needs to satisfy the constitutive equation, electric field equation, mechanical motion equation, thermal conduction equation, and thermal expansion equation; the first molybdenum electrode layer and the second molybdenum electrode layer need to satisfy the electric field equation, mechanical motion equation, thermal conduction equation, and thermal expansion equation; and the first aluminum nitride protective layer and the silicon dioxide substrate layer need to satisfy the mechanical motion equation, thermal conduction equation, and thermal expansion equation.

[0049] When performing hexahedral meshing on the geometric model of a three-dimensional thin-film bulk acoustic resonator, it is necessary to consider whether the solution of the difference equation can converge and stabilize, obtain the body center coordinates, face center coordinates, and corner coordinates of the hexahedral mesh, and then perform subsequent calculations according to the divided cell structure.

[0050] The piezoelectric constitutive equation in this embodiment is:

[0051] ;

[0052] In the formula, Represents the second-order stress tensor. This represents the second-order tensor of elastic strain. Represents the electric displacement vector. Indicates electric field strength. This represents the fourth-order tensor of the elastic constant. This represents the second-order tensor of the dielectric constant. This represents the third-order tensor of the piezoelectric stress constant. express The transpose of .

[0053] Considering the symmetry of the stress and strain tensors, the piezoelectric constitutive equation can be simplified by order reduction to: , The fourth-order tensor of the elastic constant Rewritten as a second-order tensor of elastic constants third order tensor Rewrite as the second order tensor of piezoelectric stress constant , denotes the stress tensor, denotes the elastic strain.

[0054] The electric field equation in the embodiment is:

[0055] ;

[0056] ;

[0057] In the formula, denotes the divergence, denotes the electric displacement vector, denotes the electric field intensity, denotes the electric potential, denotes the Hamiltonian operator.

[0058] The mechanical motion equation in the embodiment is:

[0059] ;

[0060] ;

[0061] ;

[0062] ;

[0063] In the formula, denotes the displacement vector, denotes the density, denotes the strain first order tensor, i.e. the total strain; denotes the stress first order tensor, denotes the time, and denotes the operator, x, y and z are three parameters of the space rectangular coordinate system.

[0064] The heat conduction equation in the prior art is: ; In the formula, denotes the density, denotes the heat capacity, denotes the thermal conductivity, denotes the heat source, denotes the time, denotes the divergence, denotes the temperature, denotes the Hamiltonian operator.

[0065] It should be appreciated that the material properties of the bulk acoustic resonator are affected by temperature, the resonant frequency of the bulk acoustic resonator is shifted, and the main electrical properties of the bulk acoustic resonator are changed. Therefore, considering the temperature conditions of the three-dimensional thin film bulk acoustic resonator geometric model in the embodiment are affected by the external environment temperature, the boundary condition should be set as a fixed boundary, so the heat source term should be zero, that is, the new heat conduction equation in the embodiment should be .

[0066] Due to the influence of external temperature, the device temperature rises, thereby causing deformation, which in turn affects the electrical properties of the device, so the influence of thermal strain needs to be considered. Therefore, the thermal-mechanical-electric coupling equation in the embodiment also needs to use the thermal expansion equation, that is, ; in the formula, represents thermal strain, represents temperature, represents time, represents the thermal expansion coefficient at time and temperature , represents the reference temperature; the strain first-order tensor and the thermal strain of the device generated after the external environment temperature is applied are calculated, and the thermal strain , the elastic strain is obtained according to the strain first-order tensor

[0067] , and then substituted into the piezoelectric constitutive equation for solving. .

[0068] It should be appreciated that the time domain finite difference method (FDTD) in the embodiment divides the calculation domain into difference grids, replaces the continuous calculation domain with a finite number of grid nodes, and continuously updates the values of field components through time iteration, which is clear and intuitive, saves storage space, is suitable for parallel computing, improves calculation efficiency, and is more conducive to processing complex problems. The accuracy of the time domain finite difference method depends only on the completeness condition. However, the finite element used in existing commercial software has other supplementary conditions to ensure convergence, and the upper bound of the discrete error of the time domain finite difference method is lower than the upper bound of the discrete error reached by the approximation theorem used in the finite element precision analysis.

[0069] In the process of solving the thermal-mechanical-electric coupling equation, the field components in the time domain finite difference (FDTD) algorithm can be continuously updated by the values of the field components around the past time point, and the differentials with respect to space and time variables are represented in difference form, which can be directly calculated through the difference equation, and also avoids the problem of matrix inversion. The difference form derivation process of the heat conduction equation is as follows: ​

[0070]

[0071] wherein i, j and k represent the coordinates of the grid nodes in the spatial rectangular coordinate system.

[0072] When the time step iteration starts, the displacement vector and the mechanical motion equation are used to obtain the strain first-order tensor Meanwhile, the temperature change is obtained from the heat conduction equation, and the generated thermal strain is obtained According to the strain first-order tensor and the thermal strain , the elastic strain is calculated; the elastic strain is brought into the Poisson equation in the electric field equation to solve the electric potential ; the electric potential and the elastic strain are brought into the piezoelectric constitutive equation to obtain the stress first-order tensor , the stress first-order tensor is brought into the mechanical motion equation to solve the displacement vector of the next time , and then the time iteration calculation is performed until the preset time is reached.

[0073] Specifically, in the process of solving the electric potential, three equations of and are solved to obtain ; and , thus .

[0074] In the process of solving the stress first-order tensor , .

[0075] The application will be described below in combination with specific embodiments.

[0076] Embodiment 1

[0077] The three-dimensional bulk acoustic wave resonator device thermal-force-electric coupling analysis method based on the FDTD algorithm is used to perform characteristic analysis on the target film bulk acoustic wave resonator, and each physical parameter in the thermal-force-electric coupling equation of the resonator under different environmental temperatures is obtained.

[0078] The displacement vector u in each physical parameter of the resonator under a certain environmental temperature is used to calculate the electric charge at different times; the electric charge at different times is used to obtain the current at different times; and the current at different times is used to obtain the current-time curve of the resonator under a certain temperature. The current-time curve of the resonator under normal temperature is as shown in Figure 2The resonator current change curve with time at different temperatures is shown in FIG. 2. Figure 3

[0079] The above merely provides the preferred but non-restrictive embodiment of the present application, and the protection scope of the present application should not be limited thereto, and any person skilled in the art should understand that any equivalent substitution or change made according to the technical solution and the inventive concept of the present application within the technical scope disclosed by the present application should be covered within the protection scope of the present application.​

Claims

1. A thermo-mechanical-electrical coupling analysis method for three-dimensional bulk acoustic resonator devices based on the FDTD algorithm, characterized in that, include: Establish a three-dimensional geometric model of a thin-film bulk acoustic resonator; The geometric model of the three-dimensional thin-film bulk acoustic resonator is subjected to parameter and boundary condition settings and hexahedral meshing to obtain the cellular structure. Based on the cellular structure, a thermo-mechanical-electric coupling equation is established; among which, the thermo-mechanical-electric coupling equation includes the piezoelectric constitutive equation, the electric field equation, the mechanical motion equation, the heat conduction equation, and the thermal expansion equation; The piezoelectric constitutive equation is: In the formula, Represents the first-order stress tensor. Indicates elastic strain. Represents the electric displacement vector. Indicates electric field strength. This represents the second-order tensor of the elastic constant. This represents the second-order tensor of the dielectric constant. This represents the second-order tensor of the piezoelectric stress constant. express Transpose of; in, In the formula, Indicates elastic strain. This represents the first-order strain tensor. Represents thermal strain; where, In the formula, Indicates thermal strain, T Indicates temperature. t Indicates time, Indicates time t and temperature T The coefficient of thermal expansion at that point, Indicates reference temperature; The finite-difference time-domain algorithm is used to represent the differentials of spatial and temporal variables in finite-difference form, and the calculations are performed directly through difference equations; once the time step iteration begins, the displacement vector... The strain first tensor is obtained from the equations of motion of mechanics. Simultaneously, the temperature change is obtained from the heat conduction equation, and the resulting thermal strain is obtained. According to the first-order strain tensor and thermal strain elastic strain is obtained ; elastic strain Substituting the Poisson equation into the electric field equation to solve for the electric potential ; to electric potential and elastic strain Substituting into the piezoelectric constitutive equation, we obtain the first-order stress tensor. The first-order stress tensor Substitute the values ​​into the equations of motion to solve for the displacement vector at the next time step. Then, the calculation is iterated over time until the preset time is reached.

2. The thermo-mechanical-electrical coupling analysis method for three-dimensional bulk acoustic resonator devices based on the FDTD algorithm according to claim 1, characterized in that, The heat conduction equation should be: ； In the formula, Indicates density, Indicates constant pressure heat capacity. Indicates thermal conductivity. Indicates time, Denotes divergence, Indicates temperature. Represents the Hamiltonian operator; in, In the formula, x, y, and z are three parameters of the spatial rectangular coordinate system.

3. The thermo-mechanical-electric coupling analysis method for three-dimensional bulk acoustic resonator devices based on the FDTD algorithm according to claim 1, characterized in that, The equation for thermal expansion is: ； In the formula, Indicates thermal strain, Indicates temperature. Indicates time, Indicates time and temperature The coefficient of thermal expansion at that point, Indicates the reference temperature.

4. The thermo-mechanical-electric coupling analysis method for three-dimensional bulk acoustic resonator devices based on the FDTD algorithm according to claim 1, characterized in that the electric field The equation is: ； In the formula, Denotes divergence, Represents the electric displacement vector. Indicates electric field strength. Represents electric potential, This represents the Hamiltonian operator.

5. The thermo-mechanical-electric coupling analysis method for three-dimensional bulk acoustic resonator devices based on the FDTD algorithm according to claim 1, characterized in that, The equations of motion are: ； ； In the formula, Represents the displacement vector. Indicates density, This represents the first-order strain tensor. Represents the first-order stress tensor. Indicates time, and Let x, y, and z represent the operators, and let z represent the three parameters of the Cartesian coordinate system.

6. The thermo-mechanical-electric coupling analysis method for three-dimensional bulk acoustic resonator devices based on the FDTD algorithm according to claim 1, characterized in that, The geometric model of the three-dimensional thin-film bulk acoustic resonator includes, from top to bottom, a first aluminum nitride protective layer, a first molybdenum electrode layer, a second aluminum nitride piezoelectric layer, a second molybdenum electrode layer, and a silicon dioxide substrate layer.

7. The thermo-mechanical-electric coupling analysis method for three-dimensional bulk acoustic resonator devices based on the FDTD algorithm according to claim 6, characterized in that, The second aluminum nitride piezoelectric layer needs to satisfy the piezoelectric constitutive equation, electric field equation, mechanical motion equation, heat conduction equation, and thermal expansion equation; The first and second molybdenum electrode layers need to satisfy the electric field equation, the mechanical motion equation, the heat conduction equation, and the thermal expansion equation. The first aluminum nitride protective layer and the silicon dioxide substrate layer need to satisfy the equations of motion, thermal conduction, and thermal expansion.

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