A method for calculating the natural frequency of contact vibration at the wafer bonding interface

Abstract

The present invention provides a method for calculating the natural frequency of contact vibration of a wafer bonding interface, including: S1. Measuring the surface topography of the wafer, calculating the fractal dimension and fractal roughness of the wafer surface according to the measured profile height distribution, and obtaining the wafer surface profile height function with nano-scale rough fractal characteristics; S2. Constructing a substrate with a smooth profile on the nano-scale, and establishing a molecular dynamics model of the wafer bonding interface with nano-scale rough fractal characteristics according to the profile height function of the wafer surface; S3. Simulating the dynamic bonding process of the wafer interface, and fitting to obtain the functional relationship between the bonding force and the normal deformation amount of the wafer; S4. Equivalent the normal contact vibration of the wafer bonding interface to a single-degree-of-freedom spring-mass system, and determine the natural frequency of contact vibration of the wafer bonding interface. The present invention takes into account the nano-scale rough fractal characteristics of the wafer surface and determines the natural frequency of the wafer bonding interface corresponding to different bonding forces.

Description

Technical Field

[0001] The present invention relates to the field of wafer-level packaging contact vibration, and specifically to a method for calculating the natural frequency of contact vibration of a wafer bonding interface. Background Art

[0002] Wafer bonding is a common wafer-level packaging technology in the microelectronics field and is widely used in high-end chip manufacturing. The actual wafer surface is not a completely smooth ideal plane, but a rough surface composed of countless rough peaks of different sizes at the nanoscale. During the wafer bonding process, the extrusion deformation between the rough peaks of the bonding interface easily causes local stress concentration in the bonding area, and the existence of phenomena such as impact and rebound results in inevitable contact vibration at the wafer interface. To effectively avoid resonance at the wafer bonding interface and improve the bonding strength of the wafer, it is necessary to consider the influence of the wafer surface topography and accurately obtain the natural frequency of contact vibration of the wafer bonding interface.

[0003] Currently, there are mainly two methods to obtain the natural frequency of the wafer bonding interface: the first is to perform modal analysis through methods such as the exciter method and the vibration table test method. Affected by human factors, experimental conditions, sample differences, etc., the experimental results have errors, and the test process is cumbersome and inefficient. The second is to establish the vibration differential equation of the wafer bonding interface from a theoretical level and then determine its natural frequency, ignoring the scale effect of the nanoscale roughness of the wafer surface and making it difficult to accurately calculate its natural frequency. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for calculating the natural frequency of contact vibration of a wafer bonding interface that considers the scale effect of the nanoscale roughness of the wafer surface.

[0005] To solve the above technical problems, the present invention adopts the following technical solutions:

[0006] A method for calculating the natural frequency of contact vibration of a wafer bonding interface includes the following steps:

[0007] S1. Use an atomic force microscope to measure the surface topography of the wafer, calculate the fractal dimension and fractal roughness of the wafer surface according to the measured contour height distribution, and obtain the wafer surface contour height function with nanoscale rough fractal characteristics;

[0008] S2. Construct a substrate with a smooth contour at the nanoscale, screen and delete the atoms on the substrate surface according to the contour height function of the wafer surface, so as to establish a molecular dynamics model of the wafer bonding interface with nanoscale rough fractal characteristics;

[0009] S3. Simulate the dynamic bonding process of the wafer interface, obtain the equivalent deformation amount of the wafer surface under different bonding forces, and fit the functional relationship between the bonding force and the normal deformation amount of the wafer;

[0010] S4. Equivalent the normal contact vibration of the wafer bonding interface to a single-degree-of-freedom spring-mass system, solve the differential equation of the nonlinear vibration of the bonding interface, and determine the natural frequency of the contact vibration of the wafer bonding interface.

[0011] Further, the step S1 includes the following steps:

[0012] Use an atomic force microscope to measure the surface profile height distribution of the wafer, calculate the autocorrelation function R(τ) of its surface according to the rough morphology of the wafer surface at the nanoscale, obtain the power spectral function S(ω) of the wafer surface by performing a Fourier transform on the autocorrelation function R(τ), use the least squares method to fit the double-logarithmic scatter plot of the power spectral function S(ω) and the spatial frequency ω, and determine the fractal dimension D and the fractal roughness G of the wafer surface according to the slope K0 and the intercept B0 of the fitting line, so as to obtain the wafer surface profile height function Z(x, y) with nanoscale rough fractal characteristics.

[0013] Further, the S2 is specifically as follows:

[0014] Construct a nanoscale smooth regular matrix in the molecular dynamics software, realize the screening and deletion of the matrix surface atoms according to the profile height function Z(x, y) of the wafer surface, obtain a wafer with a nanoscale rough fractal surface that is more in line with the actual situation, so as to establish a molecular dynamics model of the wafer bonding interface at the nanoscale.

[0015] Further, screen the atoms by determining the magnitude relationship between the profile height value z(x, y) corresponding to different positions (x, y) on the wafer surface and the z-axis coordinate value of the atoms at this position, that is, screen and delete the matrix surface atoms with z-axis coordinate values higher than the corresponding profile height value z(x, y).

[0016] Further, set the x and y directions of the molecular dynamics model to periodic boundary conditions, and set the z direction to non-periodic boundary conditions to eliminate the boundary effect; constrain the contact system through the ensemble to ensure that the simulation is carried out under macroscopic stable conditions; control the temperature of the constant temperature layer to room temperature through a correction factor to eliminate the atomic thermal motion effect. During the bonding process of the upper and lower wafers, the interatomic interaction force is described by a potential function.

[0017] Further, the S3 is specifically as follows:

[0018] Uniformly move the matrix with a nanoscale rough fractal profile to simulate the dynamic bonding process of the upper and lower wafers, and obtain the bonding force F required at different normal deformation amounts δ of the interface b , and fit to obtain the bonding force F b The power-law function relationship with the normal deformation amount δ can be expressed as:

[0019]

[0020] In the formula, the parameters k and n are both fitting coefficients, E represents the elastic modulus of the lower wafer, A0 represents the nominal contact area, and L represents the sampling length, and its value is determined according to the sampling lengths in the x and y directions of the wafer surface.

[0021] Further, the bonding force F b and the normal deformation δ are normalized to obtain the dimensionless bonding force F b * and the dimensionless normal deformation δ * , and the power-law function relationship between the dimensionless bonding force F b * and the dimensionless normal deformation δ * is obtained by fitting, which can be expressed as:

[0022]

[0023] Further, the specific content of S4 is as follows:

[0024] The normal contact vibration of the wafer bonding interface is equivalent to a single-degree-of-freedom spring-mass system. Considering the influence of the preload F0 on the wafer surface, a differential equation for the nonlinear vibration of the bonding interface is constructed, and its expression is:

[0025]

[0026]

[0027] In the formula, m represents the mass of the lower wafer, z represents the normal displacement, reflecting the change in the deformation of the wafer bonding interface, and z i represents the initial normal static displacement generated by considering the wafer under the preload F0,

[0028] Let z i = [(F0 / kEA0) 1 / n / L; then the above formula can be expressed as:

[0029] Further, the single-degree-of-freedom spring-mass motion equation is normalized by the following dimensionless variables, which can be expressed as:

[0030]

[0031] In the formula, ω i represents the natural frequency corresponding to the initial normal static displacement z i , ω i = (nKz i n-1 / m) 1 / 2 , z * represents the dimensionless normal displacement, ω* represents the dimensionless frequency, t * represents the dimensionless time.

[0032] Normalization formula The dimensionless single-degree-of-freedom spring-mass motion equation is obtained and can be expressed as:

[0033]

[0034] Solve the differential equation of the nonlinear vibration of the bonding interface to determine the natural frequency of the contact vibration of the wafer bonding interface.

[0035] Further, according to the boundary conditions z * (0) = z0 * , z * '(0) = 0, solve the dimensionless single-degree-of-freedom spring-mass motion equation to obtain the dimensionless natural frequency ω0 * , which can be expressed as:

[0036]

[0037] In the formula, α2 and α3 are parameters, α2 = (n - 1) / 2, α3 = [(n - 1)(n - 2)] / 6, z0 * represents the dimensionless initial displacement.

[0038] Compared with the prior art, the present application has the following beneficial effects:

[0039] 1. The present invention calculates the natural frequency of the contact vibration of the wafer bonding interface through a theoretical method combining numerical and analytical methods, overcoming the influences of experimental conditions, human factors, sample differences, etc., and avoiding the limitations of large experimental errors, cumbersome measurement processes, and low efficiency. 2. The present invention considers the nano-scale rough topography of the wafer surface and the influence of the preload on the natural frequency of the wafer bonding interface. Description of the Drawings

[0040] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0041] Figure 1 It is a flowchart of the method for calculating the natural frequency of the contact vibration of the wafer bonding interface of the present invention.

[0042] Figure 2 It is a schematic diagram of the calculation idea of the natural frequency of the contact vibration of the wafer bonding interface of the present invention.

[0043] Figure 3 Schematic diagram of the molecular dynamics model for the upper and lower wafer bonding process of the present invention.

[0044] Figure 4 For the dimensionless bonding force F of the wafer of the present invention b * Regarding the dimensionless normal deformation δ * Variation curve.

[0045] Figure 5 For the dimensionless natural frequency ω0 of the wafer bonding interface of the present invention * Regarding the dimensionless initial displacement z0 * Variation curve. Specific implementation manners

[0046] The technical solutions of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0047] To make the objectives, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Apparently, the described embodiments are some but not all of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0048] The present invention will be described in detail below with reference to the accompanying drawings. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0049] Embodiment: Taking the bonding interface of 12-inch silicon-silicon wafers as an example;

[0050] The present invention provides a method for calculating the contact vibration natural frequency of a wafer bonding interface, and the specific process schematic diagram is as shown in Figure 1 and Figure 2 shown. The method includes the following steps:

[0051] S1. Use an atomic force microscope to measure the surface topography of the wafer, calculate the fractal dimension and fractal roughness of the wafer surface according to the measured profile height distribution, and obtain the wafer surface profile height function with nano-scale rough fractal characteristics;

[0052] In this embodiment, the autocorrelation function R(τ) of the wafer surface is calculated based on the rough morphology of the wafer surface at the nanoscale. The power spectrum function S(ω) of the wafer surface is obtained by performing a Fourier transform on the autocorrelation function R(τ). The least squares method is used to fit the double logarithmic scatter plot of the power spectrum function S(ω) and the spatial frequency ω. The fractal dimension D and the fractal roughness G of the wafer surface are determined according to the slope K0 and the intercept B0 of the fitting line, and the wafer surface profile height function Z(x, y) with nanoscale rough fractal characteristics is obtained. The fractal dimension D determines the high and low frequency components of the rough surface (2 < D < 3); the fractal roughness G represents the height proportionality coefficient independent of the roughness frequency. Table 1 shows the morphological parameters of the wafer surface of the present invention

[0053] Table 1

[0054]

[0055] S2. Construct a substrate with a smooth profile at the nanoscale, and screen and delete the surface atoms of the substrate according to the profile height function of the wafer surface, so as to establish a molecular dynamics model of the wafer bonding interface with nanoscale rough fractal characteristics;

[0056] As Figure 3 shown, the basic size of this model is 10.86×10.86×21.72 nm. The model can be divided into 1 - rigid layer, 2 - Newton layer, 3 - constant temperature layer, and 4 - boundary layer from top to bottom. The height ratio of each simulation layer is about 8:6:1:1. The x and y directions of the model are set as periodic boundary conditions, and the z direction is set as non - periodic boundary conditions to eliminate the boundary effect; the contact system is constrained by the microcanonical ensemble (NVE) to ensure that the simulation is carried out under macroscopic stable conditions; the temperature of the constant temperature layer is controlled at room temperature, usually set at 300 K, by the velocity scaling method to eliminate the atomic thermal motion effect. During the bonding process of the upper and lower wafers, the interaction force between silicon atoms is described by the SiC.sw potential in the Stillinger - Weber potential, which can be expressed as:

[0057]

[0058] In the formula, the first summation term is the pair potential of the system, and the second summation term is the three - body potential of the system. r ij 、r ik 、r jk respectively represent the distances between atoms i, j, and k pairwise.

[0059] S3. Simulate the dynamic bonding process of the wafer interface, obtain the equivalent deformation amount of the wafer surface under different bonding forces, and fit the functional relationship between the bonding force and the normal deformation amount of the wafer;

[0060] In this embodiment, a substrate with a nano-rough fractal profile is moved along the negative z-direction at a speed V = 10 m / s to simulate the dynamic bonding process of the upper and lower wafers. The z-direction resultant force of silicon atoms in the 1-rigid layer is statistically defined as the wafer interface bonding force F b , and according to the displacement H of the 1-rigid layer z , the interface spacing d, and the cut-off distance r0, the normal deformation amount δ of the bonding interface is defined, and the bonding force F required at different normal deformation amounts δ of the interface is obtained b . Through normalization, the dimensionless bonding force F b * and the dimensionless normal deformation amount δ * are obtained, which can be expressed as:

[0061]

[0062] In the formula, E represents the elastic modulus of silicon, E = 190 GPa, A0 represents the nominal contact area, A0 = 117.94 nm 2 , and L represents the sampling length, whose value is determined according to the lengths of the simulated wafer surface in the x and y directions, L = 10.86 nm.

[0063] The dimensionless bonding force F of the wafer b * with respect to the dimensionless normal deformation amount δ * varies as shown in Figure 4 . The dimensionless bonding force F b * increases non-linearly with the increase of the dimensionless normal deformation amount δ * . The fitting result of the dimensionless bonding force F b * with respect to the dimensionless normal deformation amount δ * is shown in Table 2. The goodness of fit R 2 represents the ratio of the regression sum of squares to the total sum of squared deviations (0 ≤ R 2 ≤ 1). The closer this value is to 1, the more significant the regression effect and the better the fitting effect. The goodness of fit R 2 values in the table are all close to 1, indicating that there is a power-law function relationship between the dimensionless bonding force F b * and the dimensionless normal deformation amount δ * , which can be expressed as:

[0064]

[0065] In the formula, the parameters k and n are both fitting coefficients.

[0066] As shown in Table 2, the dimensionless bonding force F of the wafer interface of the present invention b * with respect to the dimensionless normal deformation amount δ* Fitting results

[0067] Table 2

[0068]

[0069] S4. Equivalent the normal contact vibration of the wafer bonding interface to a single-degree-of-freedom spring-mass system. Considering the influence of the preload F0 on the wafer surface, solve the differential equation of the non-linear vibration of the bonding interface to determine the natural frequency of the contact vibration of the wafer bonding interface.

[0070]

[0071]

[0072] In the formula, m represents the mass of the wafer, z represents the normal displacement, reflecting the change in the deformation of the bonding interface, z i represents the initial normal static displacement generated by considering the preload F0 borne by the wafer, z i =[(F0 / kEA0) 1 / n / L.

[0073] Let K = kEA0 / L n , Equation (5) can be expressed as:

[0074]

[0075] By normalizing the single-degree-of-freedom spring-mass motion equation with the following dimensionless variables, it can be expressed as:

[0076]

[0077] In the formula, ω i represents the natural frequency corresponding to the initial normal static displacement z i , ω i =(nKz i n-1 / m) 1 / 2 , z * represents the dimensionless normal displacement, ω * represents the dimensionless frequency, t * represents the dimensionless time.

[0078] Normalizing Equation (6) gives the dimensionless single-degree-of-freedom spring-mass motion equation, which can be expressed as:

[0079]

[0080] According to the boundary conditions z * (0)=z0 * , z *'(0) = 0, solve the dimensionless single-degree-of-freedom spring-mass motion equation (8) to obtain the dimensionless natural frequency ω0 * , which can be expressed as:

[0081]

[0082] where α2 and α3 are parameters, α2 = (n - 1) / 2, α3 = [(n - 1)(n - 2)] / 6, and z0 * represents the dimensionless initial displacement.

[0083] The dimensionless natural frequency ω0 of the wafer bonding interface * with respect to the dimensionless initial displacement z0 * variation curve, as Figure 5 shown. It can be seen that at the initial static equilibrium position z0 * = 0, the dimensionless natural frequency ω0 * reaches its maximum value; as the initial displacement z0 * deviates from the static equilibrium position of the bonding interface, the dimensionless natural frequency ω0 * begins to decrease. When the dimensionless initial displacement z0 * = 0.5, for the wafer bonding interface with a smaller roughness relative to the static equilibrium position, the dimensionless natural frequency ω0 * decreases to 0.956, and for the interface with a larger roughness, the dimensionless natural frequency ω0 * decreases to 0.737.

[0084] The above description of the preferred embodiments is relatively detailed, and it should not be considered as a limitation to the protection scope of the present invention. Under the inspiration of the present invention, those of ordinary skill in the art can also make substitutions or deformations without departing from the protection scope defined by the claims of the present invention, and all fall within the protection scope of the present invention. The scope of protection claimed by the present invention shall be subject to the appended claims.

Claims

1. A method for calculating the natural frequency of contact vibration at the wafer bonding interface, characterized in that It includes the following steps: S1. Use an atomic force microscope to measure the surface topography of the wafer, calculate the fractal dimension and fractal roughness of the wafer surface based on the measured profile height distribution, and obtain the wafer surface profile height function with nanoscale rough fractal characteristics; S2. Construct a substrate with a smooth profile at the nanoscale, screen and delete the atoms on the substrate surface according to the profile height function of the wafer surface, so as to establish a molecular dynamics model of the wafer bonding interface with nanoscale rough fractal characteristics; S3. Simulate the dynamic bonding process of the wafer interface, obtain the equivalent deformation amount of the wafer surface under different bonding forces, and fit the functional relationship between the bonding force and the normal deformation amount of the wafer; S4. Equivalently transform the normal contact vibration of the wafer bonding interface into a single-degree-of-freedom spring-mass system, solve the differential equation of the nonlinear vibration of the bonding interface, and determine the natural frequency of the contact vibration of the wafer bonding interface.

2. The method for calculating the natural frequency of contact vibration of a wafer bonding interface according to claim 1, wherein The step S1 includes the following steps: Use an atomic force microscope to measure the surface profile height distribution of the wafer, calculate the autocorrelation function R(τ) of the wafer surface according to the rough morphology of the wafer surface at the nanoscale, obtain the power spectrum function S(ω) of the wafer surface by performing a Fourier transform on the autocorrelation function R(τ), use the least squares method to fit the double-logarithmic scatter plot of the power spectrum function S(ω) and the spatial frequency ω, and determine the fractal dimension D and fractal roughness G of the wafer surface according to the slope K0 and intercept B0 of the fitting line, and obtain the wafer surface profile height function Z(x, y) with nanoscale rough fractal characteristics.

3. A method for calculating the natural frequency of contact vibration at the wafer bonding interface according to claim 1, characterized in that: The specific content of S2 is: Construct a smooth and regular substrate at the nanoscale in the molecular dynamics software, screen and delete the atoms on the substrate surface according to the profile height function Z(x, y) of the wafer surface, obtain a wafer with a nanoscale rough fractal surface that is more consistent with the actual situation, and thus establish a molecular dynamics model of the wafer bonding interface at the nanoscale.

4. A method for calculating the natural frequency of contact vibration at the wafer bonding interface according to claim 3, characterized in that: Screen the atoms by determining the magnitude relationship between the profile height value z(x, y) corresponding to different positions (x, y) on the wafer surface and the z-axis coordinate value of the atom at this position, that is, screen and delete the substrate surface atoms with z-axis coordinate values higher than the corresponding profile height value z(x, y).

5. A method for calculating the natural frequency of contact vibration of a wafer bonding interface according to claim 3, characterized in that: The x and y directions of the molecular dynamics model are set as periodic boundary conditions, and the z direction is set as non-periodic boundary conditions to eliminate the boundary effect; the contact system is constrained by the ensemble to ensure that the simulation is carried out under macroscopic stable conditions; the temperature of the constant temperature layer is controlled at room temperature by the correction factor to eliminate the atomic thermal motion effect. During the bonding process of the upper and lower wafers, the interatomic interaction force is described by the potential function.

6. A method for calculating the natural frequency of contact vibration at the wafer bonding interface according to claim 1, characterized in that: The specific content of S3 is: Move the substrate with a nanoscale rough fractal profile at a constant speed to simulate the dynamic bonding process of the upper and lower wafers, and obtain the bonding force F required at the interface with different normal deformation amounts δ b , and fit to obtain the bonding force F b The power-law function relationship between the bonding force F and the normal deformation amount δ can be expressed as: In the formula, the parameters k and n are both fitting coefficients, E represents the elastic modulus of the lower wafer, A0 represents the nominal contact area, and L represents the sampling length, and its value is determined according to the sampling length in the x and y directions of the wafer surface.

7. A method for calculating the natural frequency of contact vibration of a wafer bonding interface according to claim 6, characterized in that: Normalize the bonding force F b and the normal deformation δ to obtain the dimensionless bonding force F b * and the dimensionless normal deformation δ * , and fit to obtain the relationship between the dimensionless bonding force F b * and the dimensionless normal deformation δ * as a power-law function, which can be expressed as:

8. A method for calculating the natural frequency of contact vibration at the wafer bonding interface according to claim 1, characterized in that: The specific content of S4 is: Equivalently transform the normal contact vibration of the wafer bonding interface into a single-degree-of-freedom spring-mass system, consider the influence of the preload F0 on the wafer surface, and construct the differential equation of the nonlinear vibration of the bonding interface. Its expression is: In the formula, m represents the mass of the lower wafer, z represents the normal displacement, which reflects the change in the deformation of the wafer bonding interface, and z i It represents the initial normal static displacement of the wafer under the preload F0. Let z i = [(F0 / kEA0) 1 / n / L; Then the above equation can be expressed as:

9. A method for calculating the natural frequency of contact vibration of a wafer bonding interface according to claim 8, characterized in that: The single-degree-of-freedom spring-mass motion equation is normalized by the following dimensionless variables and can be expressed as: where ω i represents the natural frequency corresponding to the initial normal static displacement z i , ω i =(nKz i n-1 / m) 1 / 2 , z * represents the dimensionless normal displacement, ω * represents the dimensionless frequency, t * represents the dimensionless time; Normalization formula The dimensionless single-degree-of-freedom spring-mass motion equation is obtained and can be expressed as: Solve the differential equation of non-linear vibration of the bonding interface to determine the natural frequency of contact vibration of the wafer bonding interface.

10. A method for calculating the natural frequency of contact vibration at the wafer bonding interface according to claim 9, characterized in that: According to the boundary condition z * (0) = z0 * , z * '(0) = 0, solve the dimensionless single-degree-of-freedom spring-mass motion equation to obtain the dimensionless natural frequency ω0 * , which can be expressed as: where α2 and α3 are parameters, α2 = (n - 1) / 2, α3 = [(n - 1)(n - 2)] / 6, and z0 * represents the dimensionless initial displacement.

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