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Journal of Electronic Materials

Erschienen in:

Open Access 08.03.2024 | Original Research Article

A Computational Multiscale Modeling Method for Nanosilver-Sintered Joints with Stochastically Distributed Voids

verfasst von: Zhongchao Sun, Wendi Guo, Asger Bjørn Jørgensen

Erschienen in: Journal of Electronic Materials | Ausgabe 5/2024

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Abstract

A high power density is required in wide band gap power semiconductor packaging, which has led to the popularity of sintered nanosilver as an interconnecting material. However, affected by stochastically distributed voids in its microstructure, this material in practice exhibits instability leading to reduced reliability. In this paper, a computational multiscale modeling method is proposed to simulate the influence of micro-voids on macro-properties, providing an efficient tool to analyze the aforementioned problem. At the micro-scale, the three-parameter Weibull distribution of the equivalent Young’s modulus and the normal distribution of the equivalent Poisson’s ratio are captured by Monte Carlo-based finite element simulation on the reconstructed stochastic representative elements, where the density and distribution morphology of micro-voids are taken into consideration. At the macro-scale, the effect of the microscopic voids is transferred through a random sampling process to construct the multiscale model. The effectiveness and validity of the proposed method are verified through experimental case studies involving the modeling of nanosilver-sintered joints sintered at temperatures of 275°C and 300°C. In addition, the effects of the sintering temperature on the dispersion of the micro-voids, the distribution fluctuation of the constitutive parameters, and the mechanical properties are also discussed based on numerical and experimental results.

Graphical Abstract

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Introduction

Benefiting from high breakdown voltages, high operating temperatures, and high switching frequencies, wide band gap (WBG) semiconductors, such as silicon carbide (SiC) and gallium nitride (GaN), are dominating the industry of electronics packaging to ensure a high power density.^1‐3 To release the potential of WBG semiconductors adequately and reliably, traditional Pb and lead-free solders with lower melting temperatures^4,5 are being substituted by new interconnecting materials. Considering the prerequisites for bonding power chips concerning mechanical connection, electrical transportation, and heat dissipation, low-temperature^6,7 sintering nanosilver (Ag-NP) has emerged as a compelling option within various new attachment technologies,^8,9 featuring a high operating temperature of up to 961.8°C¹⁰ and a low thermal and electrical resistance of 429 W m$^{-1}$K$^{-1}$ and 1.6 µ$\Omega $ cm,¹¹ respectively.

During the sintering process, silver nanoparticles are driven by the surface energy and defective energy of the sintering paste to coalesce into a bulk solder, where mechanical pressure and an external heat source are commonly involved to obtain a denser structure. However, the applied pressure and excessive temperature may create defects and cracks on the chip and substrate,^12,13 so they are limited to some extent. Due to the inherent nature of the sintering mechanism, stochastically distributed voids inevitably exist in the microstructure of nanoparticle-sintered joints, subsequently affecting the macro-level mechanical,¹⁴ thermal,¹⁵ and electrical properties.¹⁶ By experimental methods,¹⁷ the micro-voids observed by scanning electron microscopy (SEM) with larger porosity were found to lead to a decrease in the tensile strength and bonding strength in nanosilver-sintered joints of larger joint cross-sections¹⁸ and inadequate thickness.¹⁹ In the thermal and electrical fields, a 35% increase in porosity will result in approximate 80% and 25% decreases in thermal conductivity and electrical conductivity of nanosilver-sintered joints, respectively.¹⁶ Triggering the stress singularity point,²⁰ the micro-voids are always related to the crack initiation and propagation, then leading to reliability issues.^21,22 To model the relationship between the voids and the macro-properties, material functions-based mathematical methods²³ and model-based numerical simulation methods^24‐26 have been performed. A multiphysics coupling model²⁷ of a single-chip 3.3 kV/50 A nanosilver-sintered press-pack IGBT simulated the impacts of the void distribution, number, and ratio on the electrical–thermal stress, junction temperature, on-state voltage, and the chip current distribution, while the voids have been regarded as perforated circles with a size of 0.05 mm rather than the actual morphology. The numerical model proposed by Fei et al.²⁸ showed that the equivalent thermal conductivity of porously sintered silver is affected by the void shape of the circle, ellipse, and rectangle. Another research²⁹ used focused-ion-beam tomography to obtain the real sintering structures in the silver film, and analyzed their influence on the macroscopic stress state and yield locus of sintered silver by formulating a thermodynamic consistent continuum model, but the dispersion of the stochastically distributed voids was not included.

To take a more accurate analysis of nanosilver-sintered joints, considering the effects of the porous microstructure, the real morphology has to be taken into account rather than relying on simplifications.^30‐32 At the same time, the dispersion of the voids needs to be considered to illustrate the property fluctuation under the same sintering process.³³ Facing the problem, the morphology of the stochastically distributed voids was randomly reconstructed by keeping the same dispersion characteristics in our previous research,³⁴ which can be used as the modeling input for the finite element (FE) simulation but cannot be used directly. Although, in power electronic packaging, the bonding layers usually have small geometrical dimensions (at the millimeter level),³⁵ the nanosilver-sintered joint has a 5000-times geometric difference compared to the microstructure with stochastically distributed voids (at the micron level).³⁶ If modeling the power module containing microscopic voids directly for simulation calculations, it will bring an enormous number of meshes and computational resources, despite the improved simulation accuracy.

As one popular indirect modeling method, multi-scale theory³⁷ has been used to analyze the properties of materials such as concrete,³⁸ graphene-reinforced polypropylene nanocomposites,³⁹ and duplex stainless-steel materials⁴⁰ at the macroscopic scales, by using the representative volume elements (RVEs) to maintain the disturbance from the micro-scale and without modeling all geometric details,⁴¹ which all have good agreement with the available test data. Given that nanosilver-sintered joints exhibit a similar void morphology and multi-scale feature as the above materials, it is expected that such a theory can also effectively model their properties. However, the RVE only works on periodic elements,⁴² while the real micro-morphologies in nanosilver-sintered joints are random, and the properties of the micro-porous structure will accordingly behave as a stochastic distribution. Therefore, the universal multiscale modeling pipeline does not work on nanosilver-sintered joints. To solve the problem and build a multiscale model of nanosilver-sintered joints, which is crucial to analyze the relationship between the micro-voids and macro-properties, a computational method including Monte Carlo simulation and sampling is proposed in this paper, and verified to construct the multiscale model for nanosilver-sintered joints with stochastically distributed voids.

The remaining parts of this paper are organized as follows: “Methodology” section presents detailed information on the proposed approach to model nanosilver-sintered joints from the microscale to the macroscale by the computational method. In “Experiments” introduces the manufacturing process of the samples and the experimental results of the SEM observation and shear tests. In “Results and discussion” section provides the application and validation scenarios for the proposed approach and analyzes the influence of the stochastically distributed voids. Finally, the major conclusions from this study are drawn in “Conclusions” section.

Methodology

When a nanosilver-sintered joint with stochastically distributed microscopic voids is loaded, the voids are not directly subjected to the load, but the voids’ density, size, and distribution morphology affect the generation of internal local pressure and thus the macroscopic response of the sintered joint. In the case of mechanical loads, this effect is reflected as a change in the macroscopic constitutive relationship of the sintered joint, which will show a certain statistical distribution due to the statistical characteristics of the arbitrarily distributed microscopic voids.

Overall Methodology Flow

The whole procedure of building a finite element analysis (FEA) model of nanosilver-sintered joints considering the effect of randomly distributed microscopic voids based on a computational multiscale approach is shown in Fig. 1. This proposed procedure includes the following steps:

(1)

Based on the method of characterizing and reconstructing the random void morphology of nanosilver-sintered joints, which was previously invented and published in³⁴ and presented in “Reconstruction of voids” section, the stochastic representative elements (SREs) that are able to characterize the microscopic void distribution features of the nanosilver-sintered joints were reconstructed.

(2)

Taking in the reconstructed SREs, corresponding FEA models according to the method in “Monte Carlo-based Finite element simulation” section were established and the Monte Carlo FE simulations on the micro-scale were executed to obtain the stochastic constitutive model responses from different microscopic void morphologies of the same sintered joint.

(3)

The probability distribution of the constitutive parameters calculated by simulation in the previous step were counted, the most matched statistical model was found, and the distribution pattern has been analyzed using the method developed in “Parameter fitting” section.

(4)

The macroscopic nanosilver-sintered joint in the sequence of 3D modeling, meshing, material definition, and boundary conditions application were modeled, in which the constitutive parameters of each element were sampled from the fitted models in step (3). The detailed procedure is referred to in the instructions in “Sampling and macro modeling” section. Then, the model can be solved in FEA with the ability to consider the influence of the microscopic voids on the macroscopic properties of the nanosilver-sintered joint without increasing the geometric complexity.

Reconstruction of Voids

The flow diagram of reconstructing SREs of the microstructure of nanosilver-sintered joints is shown in Fig. 2.

(1)

SEM images recording the microstructure morphology of the nanosilver-sintered joints were obtained and the stochastically distributed voids from the sintered nanosilver using an Otsu algorithm-based threshold segmentation image processing method have been distinguished. The porosity and characterization parameters matrix was extracted, which includes the void area (A), distance (l’), and deflection angle ($\alpha $) from the geometric center and the void aspect ratio ($\delta $), noted as P.

(2)

Based on the random medium (RM) theory and the porosity-based threshold segmentation algorithm, a considerable number of void morphology samples were generated with discrete autocorrelation function parameter inputs, and their characterization parameters matrices Q $_I$ extracted as in step (1) to form a void morphology database, denoted as Q.

(3)

The similarity between the matrix P and Q $_I$ was compared by using the Jensen–Shannon divergence algorithm to find the characterization parameters matrix Q $_{max}$ with the highest matching degree and its input autocorrelation parameters of the corresponding void morphology as pending reconstruction parameters.

(4)

In the similarity calculation, if the similarity was less than the specified tolerance, the input parameters were refined with reference to pending reconstruction parameters, and steps (2 and 3) repeated until the specified tolerance was satisfied, and the final reconstructed parameters obtained.

(5)

Following the same method as in step (2), void morphology samples with the same void distribution characteristics as the input nanosilver-sintered joint microstructure were established by using the final reconstructed parameters. Under the effect of the stochastic parameter used in RM theory, the samples with the same modeling parameters show different morphology features, which exhibit statistical randomness. Hence, the samples here are noted as the SREs of the nanosilver-sintered joints.

Monte Carlo-Based Finite Element (FE) Simulation

FE Modeling of SRE

The SRE created in section “Reconstruction of Voids” stores the microstructure morphology information of the nanosilver-sintered joint in the form of a binarized matrix, and each matrix cell represents a pixel in the bitmap, where 0 represents the pixel for the void and 1 represents the medium. In the geometric modeling stage in the FE simulation environment, a medium pixel in the SRE can be equated to a two-dimensional plane element.

Based on this principle, the proposed method uses an intermediate file to record the pixelated structure characteristics of the SRE and nondestructively converts the SRE from a numerical model to a geometric model. The specific process is illustrated in Fig. 3 and introduced as follows. The core code run in the Ansys Parametric Design Language (APDL) environment for SRE FE modeling is provided in Appendix A.

(1)

The morphology features of SRE in the format of ID, row, column, and feature value were extracted and marked as N, I, J, and F, respectively, and then exported as a.txt file, which can be easily recognized by ANSYS Mechanical. In this intermediate file for SRE modeling, the ID is used to record the SRE size, the row and column are used to locate the pixel points, and the feature value is used to distinguish whether the pixel characterizes a void or a medium.

(2)

The.txt file was imported into ANSYS Mechanical and the actual size of the SRE in both X and Y directions was calculated as the product of the maximum values of the row and column and the plotting scale (the actual size of 1 pixel) of the bitmap. Following the size limit, the key points used to describe all the plane elements in ANSYS geometry space were drawn, where the interval of the key points is the length of the plotting scale.

(3)

The pixel feature values in the.txt file were read, identifying the medium components, and the corresponding key points were connected according to their locations to form 2D planes. All the two-dimensional planes representing the sintered nanosilver material by the Boolean operation were connected to build the SRE geometry model.

(4)

In the meshing stage, the plotting scale was set as the quadrangular element size to guarantee its success regardless of the morphological complexity induced by stochastically distributed voids. Then, the FE model of SRE was prepared for further simulation.

Monte Carlo-Based FE Simulation

During the elastic deformation phase, the effect of microscopic stochastically distributed voids of the nanosilver-sintered joint will be reflected as the variation of the equivalent Young’s modulus and Poisson’s ratio of the SRE by using Hooke’s law to describe the constitutive relationship of the nanosilver-sintered joint.

With the FE simulation, the tensile deformation in the tensile direction and the transverse deformation in the vertical direction of the SRE subjected to the uniform tensile load can be extracted, and, according to Eqs. 1 and 2, the equivalent Young’s modulus and Poisson’s ratio can be calculated, respectively:

$$\begin{aligned} E=\frac{Pl_x}{{\Delta }l_x} \end{aligned}$$

(1)

where the X-direction is regarded as the tensile direction, E in MPa is the equivalent Young’s modulus of SRE, P in MPa represents the uniform tensile load, l $_x$ in µm is the size of SRE along the tensile direction, and $\Delta $l$_x$ in µm is the tensile deformation.

$$\begin{aligned} \nu =-\frac{\varepsilon _y}{\varepsilon _x}=-\frac{{\Delta }l_y\cdot {l_x}}{{\Delta }l_x\cdot {l_y}} \end{aligned}$$

(2)

where $\nu $ is the equivalent Poisson’s ratio of the SRE, $\varepsilon _x$ and $\varepsilon _y$ are the tensile strain and transverse strain, respectively, l$_y$ in µm is the size of SRE along the transverse direction, and $\Delta $l$_y$ in µm is the transverse deformation.

In order to analyze the effect of the randomness of the microscopic void distribution on the constitutive relationship of the nanosilver-sintered joint, the Monte Carlo-based FE simulation was conducted. By coding an APDL program displayed in Appendix B, the different SRE of the same nanosilver-sintered joint reconstructed in section “Reconstruction of Voids” were repeatedly imported to update the FE model and simulated to obtain the corresponding equivalent Young’s modulus and equivalent Poisson’s ratio, according to the method proposed in this Section.

Parameter Fitting

From the Monte Carlo-based FE simulation responses, different SREs of the same nanosilver-sintered joint will result in different equivalent Young’s modulus and Poisson’s ratio observations, which come from the same stochastic distribution but are independent of each other.

To characterize the microscopic constitutive parameter distributions of the nanosilver-sintered joint, considering the randomness of the microscopic void distribution, the proposed method first calculates the frequency distribution of the discrete constitution parameters observations obtained from the Monte Carlo-based FE simulation in section “Monte Carlo-Based Finite Element (FE) Simulation” and plots it in the form of a continuous probability density function, and then takes the commonly used statistical distributions including the normal, log-normal, exponential, Weibull, and three-parameter Weibull distributions to fit that. The distribution form with the highest acceptance probability is selected as the stochastic distribution of the constitutive parameters, and the distribution parameters are determined at the same time.

Sampling and Macro-Modeling

Considering the size difference between the micro-voids and the macro-sintered joints and to reduce the computational burden, rather than modeling all the voids in the joints, the proposed macroscopic simulation modeling method based on Monte Carlo-based FE simulation and parameters sampling was developed as follows.

In the nanosilver-sintered joint FE simulation, a macroscopic geometric model matching the actual size of the simulation object is established first, and the element size of the joint is controlled to be the same as that of the SRE in the meshing stage.

Considering the propagation of the influence of the microscopic voids on the constitutive relationship of the nanosilver-sintered joint along the scale growth, the physical parameters of each sintered joint element should be consistent with the microstructure in the macroscopic FE simulation modeling, and, considering the randomness of the microscopic void distribution, the constitutive parameters of each element of the sintered joint will be arranged as the random sampling samples of the stochastic distribution of the equivalent Young’s modulus and the equivalent Poisson’s ratio.

Based on the obtained stochastic distribution forms in Section “Parameter fitting”, the samples with the same number of elements as the nanosilver-sintered joint are generated by a random number-based direct sampling method, and the samples are assigned to each element of the nanosilver-sintered joint using the APDL program. With this, the multiscale modeling for nanosilver-sintered joints considering the influence of microscopic stochastically distributed voids on macroscopic properties is completed.

Experiments

Sample Manufacture

The manufacturing process of the nanosilver-sintered joint samples with a typical “sandwich” structure is shown in Fig. 4, and the sample components are shown in Table I. The nanosilver paste⁴³ consists of spherical and quasi-spherical Ag nanoparticles with an average particle size of 80 nm and 800 nm, respectively, and organic matter, where polyethylene glycol is used to adjust the viscosity and polyvinyl alcohol is used to prevent cracking during the sintering process. The viscosity of the nanosilver paste is about 174 Pa s, the weight percentage of silver in the paste is 83.5%, and the heat absorption temperature is 335.4°C. The Ag nanoparticles were mixed with organics by stirring to form a uniformly nano-Ag paste in the laboratory before sample manufacturing. The die material was Si, and, to enhance the interface bonding strength and avoid delamination during the test due to the material difference between Si and nanosilver, the bottom side of the die was successively sputtered with Ti and Ag. Similarly, in order to enhance the affinity between the copper substrate and the nanosilver solder paste, the upper surface of the substrate was covered with an Ag layer.

Prior to sintering, the nanosilver paste was placed in the center of the substrate by hand scraping, using a stencil mask with a thickness of 100 µm and a window size of 5$\times $5 mm. The silicon die was placed on the surface of the solder paste and subsequently sintered.

Table I

The properties of nanosilver-sintered sample components

Sample component	Die	Bonding layer	Substrate
Material	Silicon⁴⁴	Sintered nanosilver	Copper⁴⁵
Young’s modulus (MPa)	1.31$\times $10$^5$	4.0$\times $10$^4$, ⁴⁶	$1.2\times 10^5$
Poisson’s ratio	0.22	0.38⁴⁷	0.34
Density (kg/m$^3$)	2330	8600⁴³	8960
Length (mm)	5	5	10
Width (mm)	5	5	10
Thickness (mm)	1	0.1	3
Surface treatment	Ti (50 nm), Ag (50 nm)	–	Ag (50 µm)

Table II

Sintering sample list and corresponding sintering conditions

Sample ID	Sintering temperature (°C)	Sintering time (min)	Sintering pressure (MPa)	Experiment
275-1	275	15	1	SEM
275-2	275	15	1	Shear test
275-3	275	15	1	Shear test
300-1	300	15	1	SEM
300-2	300	15	1	Shear test
300-3	300	15	1	Shear test

The samples were manufactured with the pressure-assisted sintering process⁴⁸, where the sintering pressure was 1 MPa. The sintering temperature (T$_\textrm{s}$) was 275°C and 300°C, respectively, and loaded as the curve plotted in Fig. 4. During the first stage (from room temperature to 150°C), low boiling point organics (e.g., glycol and water) begin to decompose. which reduces the fluidity of the solder paste, thus preventing the die from being buried and shorted by the extruded paste. At the second stage (150–250°C), the organics of larger molecular weight (e.g., polyethylene glycol) gradually began to decompose and the solder paste weight decreases and remained stable for a period of time, enabling further curing of the paste. For the last stage, the polyvinyl alcohol decomposed, and the solder paste sintering and curing was completed with the temperature keeping a time of T$_\textrm{s}$ of 15 min.

Following the above process, 6 samples, tabulated in Table II, were prepared for SEM observation and shear testing to verify the proposed modeling method.

SEM Observation

As shown in Fig. 5, the nanosilver-sintered joint observation specimens were fabricated using the resin inlay method, ground to the center of the specimen, and polished using a diamond polishing compound with a particle size of 0.5 µm to obtain a fine sintered silver cross-section. Using a ZEISS SEM, the microscopic porous structures of the nanosilver-sintered joints sintered at T$_\textrm{s}$ of 275°C and 300°C were captured and are shown in Fig. 6.

By comparing the SEM figures, it can be found that, when the sintering temperature was increased from 275°C to 300°C, there was the phenomenon of sintering neck growth, the contact points between adjacent silver particles, inside the sintered joint. And, at the same time, the smaller stochastically distributed voids were forced to join, resulting in increasing volume and decreasing density.

Shear Test

The shear tests of samples sintered under T $_\textrm{s}$ of 275°C and 300°C were carried out by a Dage Series 4000 Bond tester, where the sample was fixed on a vacuum suction cup and shear force was applied to the silicon die by a wedge-shaped shear tool with a shear height relative to the upper surface of the substrate of 100 µm. The shear tool moved at a rate of 200 µm/s and stopped when the die was removed from the nanosilver-sintered joint. The force captured by sensors at that time was termed as the shear force and is recorded in Table III, and, based on Eq. 3, the shear strength was calculated as:

$$\begin{aligned} \tau =\frac{F}{A} \end{aligned}$$

(3)

where $\tau $ in MPa is the shear strength, F in N is the shear force, and A is the shear section area, which is equal to 25 mm$^2$.

It can be seen from the schematic of the shear section in Fig. 7 that the failure phenomenon of die breakage, the sintered layer damage, and the sintered layer peeling from the substrate may occur in the cross-section when the nanosilver-sintered joint is subjected to the shear damage, where the sintered layer damage takes the dominant percentage, so the shear test developed in this section can be used to evaluate the shear properties of the nanosilver-sintered joint. It can be concluded from the results of shear strength in Table III that the increased sintering temperature of the nanosilver-sintered joint can lead to a better capability against the shear load.

Table III

Nanosilver-sintered joint samples shear test results

Sample ID	Sintering temperature (°C)	Shear force (N)	Shear strength (MPa)
275-2	275	390.90	15.64
275-3	275	346.56	13.86
300-2	300	873.32	34.93
300-3	300	991.78	39.62

Results and Discussion

SREs Reconstruction and FE Simulation

Taking the SEM figures in Fig. 6, which demonstrate the microscopic porous structures of the nanosilver-sintered joints as the input to reconstruct the corresponding SREs, where the image size is 60 µm$\times $45 µm. The SRE size is set as 20 µm$\times $20 µm which is relatively small when compared to the sintered joint but large enough to contain the structural information. Three equal images of every sample were cut off sequentially from the centered typical sintering area in its SEM figure and shown in Fig. 8, whose porosities were listed in Table IV.

Table IV

The porosity and reconstruction parameters of sintering area

Sample ID	Sintering temperature (°C)	Area ID	Porosity	Average porosity	Reconstruction parameter ($\times $10$^3$ µm)
275-1	275°C	(a)	0.2115	0.2172	a = 338.24, b = 286.76
		(b)	0.2227
		(c)	0.2174
300-1	300°C	(d)	0.1189	0.1243	a = 500.00, b = 360.29
		(e)	0.1275
		(f)	0.1265

It can be seen that the increased sintering temperature leads to a tighter sintering structure with lower porosity. In addition, the similar porosities of three sintering areas from the same sintering sample represent a homogeneous sintered structure, thus the selected areas can be used to generate the SREs for the corresponding sintered joint.

Following the reconstruction method in section “Reconstruction of Voids” with a strict similarity tolerance of 2%, the reconstruction parameters for generating the SRE were calculated with 4 round iterations and are listed in Table IV, where a represents the autocorrelation parameter in the horizontal direction and b of the vertical direction. The larger autocorrelation parameters indicate that the increase in sintering temperature leads to larger microscopic voids as a result of the voids being squeezed and then merged by the growing sintered neck, which is consistent with the conclusions in section “SEM observation”. This phenomenon can also be observed from Fig. 8 and the two reconstructed SREs plotted in Fig. 9.

Referring to the method described in section “FE modeling of SRE”, the SREs plotted in Fig. 9 were modeled in ANSYS for further simulated calculation. The input material parameters are listed in Table I. As for meshing, the element type was chosen as PLANE182 to simulate the elastic behavior. The maximum element size was 58.8$\times $10$^{-3}$ µm, which is 1 pixel in the SRE bitmap.

To obtain the equivalent Young’s modulus and Poisson’s ratio, the tensile simulation as described in section “Monte Carlo-based FE simulation” was executed, where a 1-MPa uniform load was applied to the right boundary, and other constraint conditions are tabulated in Table V and depicted in Fig. 10, where the coupling degrees of freedom (DOF) means that the coupled nodes have identical motion.

Table V

SRE finite element simulation boundary conditions and loads

Location	Boundary condition	Object	Value
X = 0	dx = 0	Nodes	−
X = 20 µm	Coupling DOF		−
X = 20 µm	Uniform load P		1 MPa
Y = 0	dy = 0		−
Y = 20 µm	Coupling DOF		−

The simulation results are shown in Fig. 11, where the displacement ranges are $0.117\times 10^{-2}~\upmu \textrm{m}$ and $0.667\times 10^{-3}~\upmu \textrm{m}$, respectively. The displacement differences along the tensile direction prove that stochastically distributed voids lead to the inhomogeneous displacement distribution which should present a uniform gradient change in a homogeneous material; the larger the size of the voids along the displacement direction, the greater the displacement gradient. In addition, the increased porosity of the sintered joints with lower sintering temperatures aggravates the deformation of corresponding SREs and worsens the elastic tensile resistance.

Extracting the deformation of SRE in the X-direction and the Y-direction, and according to Eqs. 1 and 2, the equivalent Young’s modulus and equivalent Poisson’s ratio of SREs are calculated in Eqs. 4–7, where the subscripts stand for the sintering temperature. Compared to the input Young’s modulus and Poisson’s ratio values, 40 MPa and 0.38, respectively, the existence of stochastically distributed voids leads to a drop in the equivalent Young’s modulus and Poisson’s ratio, whose magnitudes enlarge with increasing porosity:

$$\begin{aligned}&E_{275}=\frac{Pl_{x275}}{{\Delta }l_{x275}}=\frac{1 \textrm{MPa}\times 20~ \upmu \textrm{m}}{0.117\times 10^{-2}~\upmu \textrm{m}}=17094.02~\textrm{MPa} \end{aligned}$$

(4)

$$\begin{aligned}&E_{300}=\frac{Pl_{x300}}{{\Delta }l_{x300}}=\frac{1~\textrm{MPa}\times 20~\upmu \textrm{m}}{0.667\times 10^{-3}~\upmu \textrm{m}}=29985.01~\textrm{MPa} \end{aligned}$$

(5)

$$\begin{aligned}&\nu _{275}=-\frac{{\Delta }l_{y275}}{{\Delta }l_{x275}}=\frac{0.377\times 10^{-3}~ \upmu \textrm{m}}{0.117\times 10^{-2}~\upmu \textrm{m}}=0.3230 \end{aligned}$$

(6)

$$\begin{aligned}&\nu _{300}=-\frac{{\Delta }l_{y300}}{{\Delta }l_{x300}}=\frac{0.245\times 10^{-3}~ \upmu \textrm{m}}{0.667\times 10^{-3}~ \upmu \textrm{m}}=0.3682 \end{aligned}$$

(7)

The Monte Carlo-Based FE Simulation and Parameters Fitting

To scale the effect of the stochastically distributed voids on the dispersion of the nanosilver-sinterered joint’s properties, following the method proposed in section “Monte Carlo-based FE simulation”, 50 sets of SREs were reconstructed based on the same pair of reconstruction parameters listed in Table VI and imported into the Monte Carlo-based simulation process to obtain the results of equivalent Young’s moduli and Poisson’s ratios.

The probability density function curves of the equivalent constitutive parameters of SREs with sintering temperature are plotted in Fig. 12. Compared to the solid lines representing the T$_\textrm{s}$ of 300°C, the dashed lines shift towards the left of the coordinate axis, which means that the equivalent Young’s moduli and equivalent Poisson’s ratios of SREs decrease with the increased porosity of the nanosilver-sintered joint caused by the lower sintering temperature. In addition, the distribution range of these constitutive parameters at lower sintering temperatures is larger. It can be speculated that, when the sintering temperature goes down, the microscopic voids are distributed in a more complex pattern due to the increased porosity and reduced average void size, and, as a consequence, the dispersion of parameter variations improves.

With the highest acceptance probability values of 0.9944 and 0.9966, the probability density functions of the equivalent Young’s modulus are fitted into the three-parameter Weibull distribution with the stochastic distribution form shown in Eq. 8, and the fitted stochastic distribution for the equivalent Poisson’s ratio is the normal distribution with the function shown in Eq. 9. All the fitted parameters are listed in Table VI:

$$\begin{aligned} f(x)=\frac{\beta }{\eta }\left( \frac{x-\gamma }{\eta }\right) ^{\beta -1}\exp \left[ -\left( \frac{x-\gamma }{\eta }\right) ^{\beta }\right] \end{aligned}$$

(8)

where $\eta $ is the scale parameter, $\beta $ is the shape parameter, and $\gamma $ is the location parameter.

$$\begin{aligned} f(x)=\frac{1}{\sqrt{2}\pi \sigma }\exp {\left[ -\left( \frac{x-\mu }{\sqrt{2}\sigma } \right) ^{2}\right] } \end{aligned}$$

(9)

where $\mu $ is the mean value and $\sigma $ is the standard deviation.

Table VI

The equivalent parameters fitting results

Sintering temperature (°C)	300	275	300	275
Parameter	E		${\nu }$
Distribution form	Three-parameter Weibull		Normal
$\mu $/$\eta $	1.2133	2.6106	0.3632	0.3275
$\sigma $/$\beta $	2540	4150	0.008	0.0158
$\gamma $	26,300	14,000	–	–
Acceptance probability	0.9944	0.9966	0.9670	0.9925

The Multiscale Modeling for Nanosilver Sintered Joint

Referring to the design dimensions of the test samples, a macro-scale simulation model of the nanosilver-sintered joint sample was built, as shown in Fig. 13, where the block from top to bottom is the silicon die, nanosilver-sintered joint, and copper substrate, respectively. Considering the applied pressure and decomposition of organic matter during the nanosilver-sintering process, the modeled thickness of the sintered layer was 60 µm, and other dimensions are consistent with the values in Table I.

For this 3D model, the mesh element type was set as SOILID185 to simulate the elastic behavior. Following the meshing requirements proposed in section “Sampling and macro modeling”, the element size of the sintered joint in the X, Y, and Z directions was determined as 20 µm, the same as the SRE’s dimension. The same value was also used as the element size of the silicon die and the copper substrate in the X, and Y directions to guarantee the continuous nodes at the interfaces.

The material properties of the die and the substrate and the density of the nanosilver-sintered joint are listed in Table I, while the Young’s modulus and Poisson’s ratio of each element in the sintered joint were sampled from the fitted distribution in the previous section. Benefiting from this, the sintered joint is capable of characterizing the stochastic influence from micro-voids without the need to model them physically. In this model, 187,500 pairs of Young’s modulus and Poisson’s ratio values were sampled from the three-parameter Weibull distribution and the normal distribution, respectively.

Nanosilver Sintered Joint FE Simulation and Experimental Verification

To verify the applicability of the FE model for a nanosilver-sintered joint built based on the computational multiscale modeling method proposed in this paper, a FE simulation to get the shear strength as the experiment in Sect. 3.3 was developed as follows.

Referring to the shear test conditions, the lower surface and the two sides along the Y-axis direction of the substrate were set as completely fixed constraints to simulate the fixture offered by a vacuum suction cup. A uniform load was applied on a side surface of the die parallel to the X–Z plane with a coordinate of $Y = -2.5$ µm and the direction is along the positive Y-axis., where the value of the load is calculated from the critical shear force measured in the shear test from:

$$\begin{aligned} P_\textrm{shear}=F/A_\textrm{cross} \end{aligned}$$

(10)

where P $_\textrm{shear}$ in MPa is the uniform load used in the simulation, F in N is the shear force measured in the experiment and recorded in Table III, and A $_\textrm{cross}$ represents the area of surface that the shear load P_shear is acting on, whose value is 300 µm$^2$.

In addition, the silicon die was set as a completely rigid body by coupling all the DOF of all its nodes, without considering the fracture of the silicon die during shearing. The simulation boundary constraints and load conditions are shown in Fig. 14.

The simulation results of the stresses in the Y–Z plane of the nanosilver-sintered joint model corresponding to sample 275-2 and the zoomed bonding layer are shown in Fig. 15. It can be seen that the maximum shear stress in the nanosilver-sintered joint occurs at the edge of the sintered layer with the greatest deformation during the shear process. For other samples, the stress distribution patterns are analogous. In this process, the shear deformation occurred in the Y–Z plane, so with reference to the factors of loading position, loading direction, and the shear strength calculation method, the average shear stress response in the Y–Z direction of the sintered layer nodes can be extracted as the simulated shear strength and compared with the experimental results from Sect. 3.3, and to ⁴⁹ and⁵⁰ where the samples are also sintered under 1 MPa, at 275°C and 300°C, respectively, as plotted in Fig. 16.

It can be seen that, with the increase of the sintering temperature, the shear test results and the FE simulation results based on the computational multiscale modeling method consistently showed the trend of increasing shear strength of the joints. In addition, the simulation results are fairly consistent with the shear test results, although there was a certain error. For the simulation, the error may have come from the relatively small simulation input of Young’s modulus, which was obtained from the nanoindentation test on the non-fully dense specimen.

Based on the classical double-sphere model⁵¹ and Coble’s tetrahedron model⁵² describing the sintering process of material particles, elevating the sintering temperature effectively promotes the sintering process and reduces the porosity of the sintered body, creating more sintering necks and enriching the grain boundaries. The presence of grain boundaries at room temperature can hinder the movement of dislocations, thus increasing the material’s resistance to plastic deformation. Therefore, for nanosilver-sintered joints, increasing the sintering temperature can enhance the shear strength of the joints, consistent with the results of the FE simulations conducted based on the multi-scale model proposed in this paper. Furthermore, for nanosilver-sintered materials, strength and stiffness generally have a proportional relationship. Hence, nanosilver-sintered joints obtained at higher sintering temperatures have greater resistance to elastic deformation, and a higher Young’s modulus, which shows consistent trends with the FE simulation results of the nanosilver-sintered joint SRVE in microscale.

The above analysis demonstrates that the simulation of nanosilver-sintered joints based on the computational multi-scale modeling method proposed in this study can properly predict the constitutive parameter distributions and the mechanical properties of sintered joints, and the model can achieve more accurate prediction when the accuracy of the input parameters is highly guaranteed.

Conclusions

A computational multiscale modeling method for nanosilver-sintered joints with stochastically distributed voids is proposed in this paper. Through the finite element (FE) simulation on the micro-scale porous structure and the macro-scale sintered samples with experimental verification, the following conclusions can be drawn:

(1)

A multiscale modeling method for nanosilver-sintered joints has been developed in 4 steps, including reconstruction of the stochastic representative elements (SREs) for micro-porous morphology, Monte Carlo-based FE simulation on SREs, distribution fitting for constitutive parameters, and stochastic sampling for macro-modeling. This proposed method makes it possible to analyze the influence of the micro-voids on the macro-properties in a simulation without modeling the micro-details in the simulation model.

(2)

On the microscale, an approach to convert the morphology information of SRE into an operational FE model is proposed where the SREs are reconstructed based on the characteristics of the stochastically distributed voids. The tensile simulation on the FE model shows that the stochastically distributed voids lead to an inhomogeneous displacement distribution, and the increased porosity of the sintered joints with lower sintering temperature aggravates the deformation of corresponding SREs and worsens the elastic tensile resistance.

(3)

The distribution of equivalent constitutive parameters has been captured from the Monte Carlo-based FE simulation and parameter fitting and used as the link to transfer the effect of the microscopic voids to the macroscopic scale of the sintered joint. Influenced by the stochastic distribution of the voids, the equivalent Young’s modulus shows a trend of three-parameter Weibull distribution and the equivalent Poisson’s ration has a normal distribution. When the sintering temperature goes down, the microscopic voids are distributed in a more complex pattern due to the increased porosity and reduced average void size, which were observed in the experimental SEM images and Monte Carlo-based simulation models, then the dispersion of parameter variations improves.

(4)

The proposed modeling method has been used to simulate the shear process of the nanosilver-sintered samples, verified by the shear test, where the simulated and experimental results of the shear strength show acceptable agreement, which shows the applicability of the proposed method. Both the simulations and the experiments show that increasing the sintering temperature can improve the ability of nanosilver-sintered joints to resist shear damage, which is beneficial to reduce the possibility of thermal fatigue damage in the solder layer of electronic packaging structures and to improve the inherent reliability level of nanosilver-sintered materials.

Acknowledgements

The authors would like to thank Associate Professor Cheng Qian and Associate Professor Hongqiang Zhang for their kind help in method development and experiment, respectively.

Declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Ethical Approval

This research did not contain any studies involving animal or human participants, nor did it take place on any private or protected areas.

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Appendix A: APDL Code for Building SRE FE Model

The core code run in the ANSYS APDL environment for building SRE FE model is shown below:

Appendix B: APDL Code for SRE FE Simulation

The core code run in the ANSYS APDL environment for Monte Carlo-based FE simulation of SREs is shown below:

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Titel: A Computational Multiscale Modeling Method for Nanosilver-Sintered Joints with Stochastically Distributed Voids
verfasst von: Zhongchao Sun
Wendi Guo
Asger Bjørn Jørgensen
Publikationsdatum: 08.03.2024
Verlag: Springer US
Erschienen in: Journal of Electronic Materials / Ausgabe 5/2024
Print ISSN: 0361-5235
Elektronische ISSN: 1543-186X
DOI: https://doi.org/10.1007/s11664-024-10960-x

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Sintering temperature (°C)	300	275	300	275
Parameter	E		\({\nu }\)
Distribution form	Three-parameter Weibull		Normal
\(\mu \)/\(\eta \)	1.2133	2.6106	0.3632	0.3275
\(\sigma \)/\(\beta \)	2540	4150	0.008	0.0158
\(\gamma \)	26,300	14,000	–	–
Acceptance probability	0.9944	0.9966	0.9670	0.9925

Springer Professional

Abstract

Graphical Abstract

Publisher's Note

Introduction

Methodology

Overall Methodology Flow

Reconstruction of Voids

Monte Carlo-Based Finite Element (FE) Simulation

FE Modeling of SRE

Monte Carlo-Based FE Simulation

Parameter Fitting

Sampling and Macro-Modeling

Experiments

Sample Manufacture

SEM Observation

Shear Test

Results and Discussion

SREs Reconstruction and FE Simulation

The Monte Carlo-Based FE Simulation and Parameters Fitting

The Multiscale Modeling for Nanosilver Sintered Joint

Nanosilver Sintered Joint FE Simulation and Experimental Verification

Conclusions

Acknowledgements

Declarations

Conflicts of interest

Ethical Approval

Publisher's Note

Appendix A: APDL Code for Building SRE FE Model

Appendix B: APDL Code for SRE FE Simulation

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