Numerical and Mechanical Analysis of Direct Wafer Bonding Considering Non-Uniform Impurity Particle Distributions

Abstract

Direct wafer bonding allows polished semiconductor wafers to be joined together without the use of a binder. It has a wide range of applications in integrated circuit fabrication, micro-electro-mechanical systems (MEMS) packaging and multifunctional chip integration. Chip deflection and strain energy can be used to assess the bonding quality, and impurities have an important effect on the bonding quality. In this paper, a mathematical model and a finite element model of wafer bonding are established. The effects of different impurity distributions (Cluster, Complex, Face, Line) on the bonding quality of wafers are investigated, and the results show that the curvature and thickness of the wafer as well as the distribution of the impurity particles jointly determine the strain energy of the wafer under a certain pressure. Among them, the impurity particle surface distribution has the greatest influence on the wafer bonding quality. Finite element simulations verified the correctness of the proposed model. This work provides a theoretical basis for studying the effect of impurity distribution on wafer bonding performance.

Introduction

Direct wafer bonding is defined as a bonding procedure involving two clean, highly polished wafers, with no intermediate layer used¹. Once the two polished wafers have undergone cleaning and surface activation, they are brought into contact and local attraction, adhesion, or bonding occurs via van der Waals forces or hydrogen bonds (i.e., bonds formed between hydrogen atoms and polarized atoms such as oxygen). Nevertheless, as a result of environmental and other factors, impurity particles may be present in the bonding region, hindering the wafer bonding process. The question of how to quantify the influence of impurity particles on bonding quality has emerged as a critical issue in the effort to enhance bonding efficiency and quality.

Currently, there are numerous studies on direct wafer bonding. From the perspective of molecular dynamics, Xueyi Duan et al. studied the direct bonding of Si/InP². Nagai Sho et al. investigated the influence of the acidity and alkalinity of surface hydroxyl groups on the low-temperature bonding of silica and metal oxides³. From the perspectives of chemistry and materials, Xiaocun Wang et al. proposed a method for direct Cu-Cu bonding in air⁴. Ye Li et al. studied the optimized wafer—level direct bonding of superconducting NbN for 3D chip integration⁵. Fengwen Mu et al. achieved direct GaN-Si wafer bonding at room temperature for the transfer of thin GaN devices after epitaxial lift-off⁶. Takashi Matsumae et al. studied the direct low-temperature bonding of diamond (100) on silicon wafers under atmospheric conditions⁷. Yang Xu et al. studied the direct bonding of Ga₂O₃-SiC at room temperature⁸. From the perspective of mechanical behavior during the bonding process, Nathan Ip et al. used a solid-fluid mechanics coupling simulation model to study wafer bonding dynamics^9,10. Kyeongbin Lim studied the attractive forces and micro-roughness at the wafer bonding interface through molecular dynamics simulation¹¹. E. Navarro et al. established a wafer bonding dynamics model and solved it numerically¹². F. Rieutord et al. established a theoretical model to describe the wafer deformation profile during bonding or the dependence of the velocity on the gas viscosity, pressure, and wafer thickness¹³. The bonding area can be predicted by the wafer geometry, elastic properties, and adhesion work^14,15. The above-mentioned studies did not consider the influence of voids during the bonding process. In actual processes, voids often form at the bonding interface due to wafer protrusions, impurity particles, etc.^16,17,18,19. Shizhao Wang et al. studied the influence of copper protrusions on wafer bonding²⁰. C. S. Tan et al. observed the pore formation between bonded copper interfaces and gave the reasons for pore formation²¹. Hung-Che Liu et al. studied the evolution of interface voids under different annealing conditions for Cu to Cu bonding²². Hyoeun Kim et al. studied the optimization of pores during the bonding process from the process perspective²³. Hsiang-Hou Tseng et al. oriented silver on nanotwinned copper microbumps for metal direct bonding and studied the change of pores generated during the deposition process with bonding conditions (temperature and pressure)²⁴. C. H. Huang et al. studied the influence of wafer surface roughness on pores²⁵. Tung-Yen Lai et al. studied the evolution and healing of pores during the bonding process at different temperatures by introducing artificial voids²⁶. Tomoya Iwata et al. evaluated the water stress corrosion at the wafer bonding interface and revealed that water stress corrosion was caused by moisture in the atmosphere, interface voids, and residual moisture²⁷. Sung-Hyun Oh et al. studied the influence of local voids on Cu-Cu bonding²⁸. Feixiang Tang et al. studied the impact of impurity particles on wafer bonding quality and proposed a theoretical model²⁹.

In this paper, by introducing impurity particles, the effects of impurity particles with different distributions on bonding quality were studied, and the corresponding analytical expressions were given. Moreover, the analytical results were compared and verified through the finite element method.

Mathematical modeling and numerical simulation

Figure 1 provides a schematic representation of the wafer bonding process under a uniformly distributed load P. The upper wafer’s geometry is characterized by its curvature k_i, thickness H, and radius R. It is assumed that the upper wafer is isotropic with its elastic behavior defined by the elastic modulus E and Poisson’s ratio v. Wafer warpage is characterized by wafer bow, whose attributes include the height difference δ between the wafer’s center and edge, as well as the curvature k_i. Usually, the wafer’s height difference is less than its thickness, with δ ≤ 1/3H. As illustrated in Fig. 1(b), the contact region can be split into two regions: Region I and Region II. The bonding process between the two wafers begins at the center. It is assumed that the critical contact position spreads axially symmetrically toward the outside, with the contact radius denoted as r_p. When the critical contact radius moves outward, the two wafers deform to attain a uniform curvature k_f. Once wafer bonding is complete, the upper wafer aligns with the curved surface of the fixture at the bonding point—specifically, Region I. In this situation, the final curvature k_f of the bonded wafer is only associated with the fixture’s shape, i.e., the lower wafer. For this reason, the lower wafer is assumed to be a rigid body without warpage (k_f  = 0). In practical applications, the wafer bow δ is far smaller than the wafer radius R, allowing the use of the small-angle approximation³⁰. Thus, during the bonding process, only vertical deflection is considered, while horizontal deflection is neglected. As shown in Fig. 1(a), tungsten particles W(r_i, θ_i) are introduced to hinder wafer bonding in Region I. In Region II, since the upper and lower wafers have not yet bonded, the particles W in this region have no impact on the bonding process. As depicted in Fig. 1(d), within Region I, the presence of particle W(r_i, θ_i) means the upper and lower wafers do not deform to the uniform curvature k_f. Instead, voids and a series of void tails are generated due to the particle’s obstruction. It is assumed that each void has a radius c_i, which is linearly related to the radial position of particle W³¹. Since the voids at r_i are the primary factors affecting wafer bonding, the influence of void tails is ignored. Additionally, it is assumed that the deformation shape of the upper wafer at r_i is described by the shape function N(r, θ_i).

Fig. 1: The model of direct wafer bonding with normal pressure and particles.

a 3D model before bonding. b 3D model after bonding. c Mechanical model before bonding. d Mechanical model after bonding²⁹

The core mechanism underlying direct wafer bonding at room temperature resides in the short-range molecular and atomic forces on the wafer surfaces, such as Van der Waals’ force and hydrogen bonding. The bonding ability of the wafers is assessed through the adhesive energy Γ, which stands for the energy needed per unit bonding area of the wafers³². The adhesive energy can be formulated with respect to the surface energies γ₁ and γ₂ of the wafers, along with the interface energy γ₁₂, via the expression Γ = γ₁ + γ₂ - γ₁₂. When these two surfaces come together to form a bonding interface, the net free energy reduces. Based on the criteria for direct wafer bonding, the conditions for achieving wafer bonding at room temperature are as follows:

$$U-\Gamma * A < 0$$

(1)

where U stands for the elastic strain energy of the upper wafer and A denotes the bonding area (A = π*r_p²).

Thus, in accordance with Eq. (1), the actual bonding condition can be expressed as

$$\frac{U}{\pi * {r}_{p}^{2}} < \Gamma \,$$

(2)

Given that the geometric properties of the upper wafer are determined by the curvature k_i, thickness H and radius R, the form of the upper wafer can be described as

$${w}_{i}=\frac{1}{2}{k}_{i}{r}^{2}$$

(3)

where r stands for the radial position of the wafer.

Given that the wafers possess a shared curvature k_f once the wafer bonding is completed, the post-bonding shape of the wafer can be formulated as

$${w}_{f}=\frac{1}{2}{k}_{f}{r}^{2}$$

(4)

In Region I, owing to the existence of particle W, the deformation at the location of particle W after bonding does not match the description in Eq. (4). It is postulated that this deformation can be denoted by the shape function N(r, θ_i).

$$N(r,{\theta }_{i})=\left\{\begin{array}{c}k{\left(r-{r}_{i}+{c}_{i}\right)}^{2}\,{r}_{i}-{c}_{i},{r}_{i}\\ k{\left(r-{r}_{i}-{c}_{i}\right)}^{2}\,{r}_{i},{r}_{i}+{c}_{i}\end{array}\right.$$

(5)

where θ_i stands for the particle's circumferential location, r_i denotes the radial position of the particle, c_i represents the void radius produced by the particle, k is the coefficient of the shape function N(r, θ_i), which is 2R_i/c_i². R_i denotes the particle radius.

Thus, the deflection of the wafer within the bonded area can be formulated as²⁹:

$$w(r,{\theta }_{i})=\left\{\begin{array}{l}\frac{1}{2}\left({k}_{i}-{k}_{f}\right){r}^{2}\qquad\qquad\qquad\qquad\,\,\,\,{0}{,r}_{i}-{c}_{i}\\ \frac{1}{2}\left({k}_{i}-{k}_{f}\right){r}^{2}-k{\left(r-{r}_{i}+{c}_{i}\right)}^{2}\quad\,{r}_{i}-{c}_{i},{r}_{i}\\ \frac{1}{2}\left({k}_{i}-{k}_{f}\right){r}^{2}-k{\left(r-{r}_{i}-{c}_{i}\right)}^{2}\quad\,{r}_{i},{r}_{i}+{c}_{i}\\ \frac{1}{2}\left({k}_{i}-{k}_{f}\right){r}^{2}\qquad\qquad\qquad\qquad\,\,\,\,{r}_{i}+{c}_{i},{r}_{p}\end{array}\right.$$

(6)

Within the unbonded area—specifically region II—physical obstacles exert no influence, and this segment of the wafer may be regarded as a circular plate. Based on the thin plate theory, the deformation process is presumed to involve pure bending. Since the pressure acting on the wafer is evenly distributed, the governing equation can thus be expressed as

$$D\left(\frac{{d}^{2}}{d{r}^{2}}+\frac{1}{r}\frac{d}{{dr}}\right)\left(\frac{{d}^{2}{w}_{{{II}}}}{d{r}^{2}}+\frac{1}{r}\frac{d{w}_{{II}}}{{dr}}\right)=P$$

(7)

The rigidity of a wafer, designated as D, is determined by the elastic modulus E, Poisson's ratio v and the thickness H of the wafer. From a mathematical standpoint, this can be formulated as

$$D=\frac{E{H}^{3}}{12(1-{v}^{2})}$$

(8)

the solution of Eq. (7) can be expressed as ref.²⁹:

$${w}_{{\rm{I}}}={w}_{{sp}}+{f_{1}}({r}_{p},P){ln}(r)+{f_{2}}({r}_{p},P){r}^{2}{{ln}}(r)+{f_{3}}({r}_{p},P){r}^{2}+{f_{4}}({r}_{p},P)$$

(9)

this term w_sp stands for the particular solution, which can be written as: w_sp = r⁴P/64D.

At the position r = r_p, the boundary condition can be formulated as:

$${w}_{{\rm{{\rm I}}}}\left|r={r}_{p}\right.={w}_{{\rm{{\rm I}}}{\rm{{\rm I}}}}\left|r={r}_{p}\right.$$

(10)

$$\frac{d{w}_{{\rm{{\rm I}}}}}{{dr}}\left|r={r}_{p}\right.=\frac{d{w}_{{\rm{{\rm I}}}{\rm{{\rm I}}}}}{{dr}}\left|r={r}_{p}\right.$$

(11)

At the position r = R, the boundary condition can be formulated as:

$${M}_{r}\left|r=R=-D\left(\frac{{d}^{2}{w}_{{\rm{II}}}}{d{r}^{2}}+\frac{v}{r}\frac{d{w}_{{\rm{II}}}}{{dr}}\right)\right.=0$$

(12)

$${Q}_{r}\left|r=R=-D\frac{d({\nabla }^{2}{w}_{{\rm{I}}})}{{dr}}\right.=0$$

(13)

Since the model in this paper is characterized by central symmetry, consequently

$${{\nabla}}^{2}{w}_{{\rm{II}}}=\frac{{\partial }^{2}{w}_{{\rm{II}}}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial {w}_{{\rm{II}}}}{\partial r}$$

(14)

Through substituting the boundary conditions, the general solution may be formulated as²⁹:

$$\begin{array}{l}{f}_{1}({r}_{p},P)=-\displaystyle\frac{{R}^{2}{r}_{p}^{2}(v+1)}{E{H}^{3}[({r}_{p}^{2}-{R}^{2})v-{r}_{p}^{2}-{R}^{2}]}\left\{E{H}^{3}({k}_{i}-{k}_{f})\right.\\\qquad\qquad\quad+\frac{(3-3v)[({R}^{2}-{r}_{p}^{2})v-{r}_{p}^{2}-{R}^{2}]P}{4}\\\qquad\qquad\quad\left.+\,3P{R}^{2}(v+1)(v-1)[{\mathrm{ln}}(R)-\,{\mathrm{ln}}({r}_{p})]\right\}\end{array}$$

(15. a)

$${f}_{2}({r}_{p},P)=\frac{3P{R}^{2}({v}^{2}-1)}{2E{H}^{3}}$$

(15. b)

$$\begin{array}{l}{f}_{3}({r}_{p},P)=\displaystyle\frac{v-1}{E{H}^{3}[2(-v-1){R}^{2}+2(v-1){r}_{p}^{2}]}\left\{3P{R}^{4}{(v+1)}^{2}{{ln}}(R)\right.\\\qquad\qquad\quad-\,3{r}_{p}^{2}P{R}^{2}(v-1)(v+1){{ln}}({r}_{p})+\,E{H}^{3}{r}_{p}^{2}({k}_{i}-{k}_{f})\\\qquad\qquad\quad\left.+\,\frac{[3(v+3){R}^{4}-6{r}_{p}^{2}(v-1){R}^{2}+3(v-1){r}_{p}^{4}](v+1)P}{4}\right\}\end{array}$$

(15. c)

$$\begin{array}{l}{f}_{4}({r}_{p},P)=\displaystyle\frac{{r}_{p}^{2}(v+1)}{E{H}^{3}[(-v-1){R}^{2}+(v-1){r}_{p}^{2}]}\left\{(-3{v}^{2}+3)P{R}^{4}{\mathrm{ln}}^{2}({r}_{p})\right.\\ \qquad\qquad\quad+\,{R}^{2}\,{\mathrm{ln}}({r}_{p})\left[\begin{array}{c}(3{v}^{2}-3)P{R}^{2}\,{\mathrm{ln}}(R)+\displaystyle\frac{3P(v+3){R}^{2}(v-1)}{4}\\ +\left(\displaystyle\frac{3{r}_{p}^{2}{v}^{2}}{4}-\frac{3{r}_{p}^{2}}{4}\right)P+E{H}^{3}\left({k}_{i}-{k}_{f}\right)\end{array}\right]\\ \qquad\qquad\quad-\,\displaystyle\frac{3P{R}^{4}(v-1)(v+1){\mathrm{ln}}(R)}{2}-\displaystyle\frac{3P(v+3){R}^{4}(v-1)}{8}\\\qquad\qquad\quad\left.+\left[\displaystyle\frac{(9v-9){r}_{p}^{2}P}{16}\left(v-\displaystyle\frac{5}{3}\right)-\displaystyle\frac{E{H}^{3}({k}_{i}-{k}_{f})}{2}\right]{R}^{2}-\displaystyle\frac{3P{(v-1)}^{2}{r}_{p}^{4}}{16}\right\}\end{array}$$

(15. d)

The wafer's strain energy may be formulated as the sum of strain energies in Region I and Region II, this relationship can be denoted as²⁹:

$$\begin{array}{l}\displaystyle{U}({r}_{p},P)={U}_{{\rm{I}}}({r}_{p},P)+{U}_{{\rm{I}}{\rm{I}}}({r}_{p},P)\\\qquad\qquad=\displaystyle{D}{\int_{0}^{2\pi }}\frac{1}{2}{\int_{0}^{{r}_{p}}}\left[{\left(\frac{{d}^{2}{w}_{{\rm{I}}}}{d{r}^{2}}+\frac{1}{r}\frac{d{w}_{{\rm{I}}}}{{dr}}\right)}^{2}-\frac{2(1-v)}{r}\frac{d{w}_{{\rm{I}}}}{{dr}}\frac{{d}^{2}{w}_{{\rm{I}}}}{d{r}^{2}}\right]{rdrd}\theta \\\qquad\qquad\quad+\,\displaystyle{D}{\int_{0}^{2\pi}}\frac{1}{2}{\int_{{r}_{p}}^{R}}\left[{\left(\frac{{d}^{2}{w}_{{\rm{I}}{\rm{I}}}}{d{r}^{2}}+\frac{1}{r}\frac{d{w}_{{\rm{I}}{\rm{I}}}}{{dr}}\right)}^{2}-\frac{2(1-v)}{r}\frac{d{w}_{{\rm{I}}{\rm{I}}}}{{dr}}\frac{{d}^{2}{w}_{{\rm{I}}{\rm{I}}}}{d{r}^{2}}\right]{rdrd}\theta\end{array}$$

(16)

For impurity particles at different circumferential positions θ_i, the strain energy U_I of their bonding regions can be respectively expressed as:

$${U}_{{\rm{I}}}({r}_{p},P)=\mathop{\sum }\limits_{i=1}^{n}\frac{D{R}_{i}}{{r}_{i}}\left\{\begin{array}{c}4{k}^{2}\left[{({r}_{i}-{c}_{i})}^{2}{{ln}}\left(\frac{{r}_{i}}{{r}_{i}-{c}_{i}}\right)+{({r}_{i}+{c}_{i})}^{2}{{ln}}\left(\frac{{r}_{i}+{c}_{i}}{{r}_{i}}\right)\right]+\\ 4(1+v)({k}_{f}-{k}_{i})(2k+{k}_{f}-{k}_{i}){r}_{i}{c}_{i}+(1+v){({k}_{i}-{k}_{f})}^{2}({r}_{p}^{2}-4{r}_{i}{c}_{i})\\ -(1+v){({k}_{i}-{k}_{f})}^{2}{r}_{p}^{2}\end{array}\right\}+D\pi (1+v){({k}_{i}-{k}_{f})}^{2}{r}_{p}^{2}$$

(17. a)

When impurity particles at the same circumferential positions θ_i, the strain energy U_I of their bonding regions can be respectively expressed as:

$${U}_{{\rm {I}}}({r}_{p},P)=\mathop{\sum }\limits_{i=1}^{n}\frac{D{R}_{i}}{{r}_{i}}\left\{\begin{array}{c}4{k}^{2}\left[{({r}_{i}-{c}_{i})}^{2}{{ln}}\left(\frac{{r}_{i}}{{r}_{i}-{c}_{i}}\right)+{({r}_{i}+{c}_{i})}^{2}{{ln}}\left(\frac{{r}_{i}+{c}_{i}}{{r}_{i}}\right)\right]+\\ 4(1+v)({k}_{f}-{k}_{i})(2k+{k}_{f}-{k}_{i}){r}_{i}{c}_{i}+(1+v){({k}_{i}-{k}_{f})}^{2}({r}_{p}^{2}-4{r}_{i}{c}_{i})\end{array}\right\}+D\left(\pi -\frac{{R}_{i}}{{r}_{i}}\right)(1+v){({k}_{i}-{k}_{f})}^{2}{r}_{p}^{2}$$

(17. b)

where R_i represents the radius of the particles, n represents the number of particles located at the same radial position, r_i represents the radial position of the particles, and c_i represents the void radius generated by the particles.

Since the particles do not influence Region II, the strain energy in Region II can be represented as²⁹:

$$\begin{array}{l}{{\rm{U}}}_{{\rm{II}}}=\frac{\pi }{12E{H}^{3}((v-1){r}_{p}^{2}-(v+1){R}^{2})}\left\{-18{{\rm{P}}}^{2}{r}_{p}^{2}{R}^{6}{\mathrm{ln}}^{2}({r}_{p})(v-1){(v+1)}^{2}+9{r}_{p}^{2}{R}^{4}{{\rm{P}}}^{2}\,{\mathrm{ln}}({r}_{p})({v}^{2}-1)\left[4{R}^{2}\,{\mathrm{ln}}(R)(v+1)\right.\right.\\\qquad\quad\left.+\,{r}_{p}^{2}(v+1)-{R}^{2}(v-1)\right]-18{{\rm{P}}}^{2}{r}_{p}^{2}{R}^{6}{\mathrm{ln}}^{2}(R)(v-1){(v+1)}^{2}-9{r}_{p}^{2}{R}^{4}{{\rm{P}}}^{2}\,{\mathrm{ln}}(R)({v}^{2}-1)[{r}_{p}^{2}(v+1)-{R}^{2}(v-1)]\\\qquad\quad\left.+\,({r}_{p}^{2}-{R}^{2})\left[\frac{3{{\rm{P}}}^{2}{r}_{p}^{6}(v+1){(v-1)}^{2}}{8}-\frac{3{{\rm{P}}}^{2}{r}_{p}^{4}{R}^{2}({v}^{2}-1)(9v-1)}{8}+{r}_{p}^{2}\left(\frac{45{{\rm{P}}}^{2}{R}^{4}({v}^{2}-1)}{8}\left(v-\frac{11}{15}\right)+{E}^{2}{H}^{6}{({k}_{i}-{k}_{f})}^{2}\right)-\frac{21{{\rm{P}}}^{2}{R}^{6}({v}^{2}-1)}{8}\left(v+\frac{13}{7}\right)\right]\right\}\end{array}$$

(18)

Parameters employed in this research encompass the elastic modulus E = 149 GPa, Poisson's ratio v = 0.18, wafer thickness H = 0.675 mm, wafer radius R = 75 mm and particle radius R_i = 2.5 mm. Adhesion energy prior to annealing typically ranges from 0.01-0.1 J/m². The wafer model built in this study features a thickness of several hundred micrometers; if thinned down to several tens of micrometers, the impact of the scale effect must be taken into account^33,34.

In this paper, tungsten particles W are used as impurity particles to study the influence of impurity particles with different distributions on the strain energy when the wafers are completely bonded. For particles W(r_i, θ_i), when their radial positions r_i are different but the circumferential positions θ_i are the same, the deflection curve of the bonding region is shown in Fig. 2(a). When both r_i and θ_i are different, the influence of each particle on the bonding is independent. The deflection curve of the bonding region for a single particle is shown in Fig. 2(b). It can be observed that at r_i, due to the obstruction of particle W, the deflection of the upper wafer decreases slightly, and the decrease amplitude is equal to the shape function N(r_i, θ_i) assumed in Eq. (5) of this paper.

Fig. 2: Wafer Deflection Curves.

a Particles have different radial positions r_i but the same circumferential position θ_i. b Particles have different radial positions r_i and different circumferential positions θ_I²⁹

As shown in Fig. 3, this paper studies the influence of four particle distributions on wafer bonding. Figure 3 (a) is the Cluster distribution, in which particles are distributed circumferentially around the central axis. Figure 3 (b) is the Complex distribution, with particles randomly distributed on the wafer. Figure 3 (c) represents the Face distribution, where particles have the same radial position r_i but different circumferential positions θ_i. Figure 3 (d) is the Line distribution, where particles have different radial positions r_i but the same circumferential position θ_i.

Fig. 3: Schematic diagrams of particle distributions.

(a) Cluster (b) Complex (c) Face (d) Line

As shown in Fig. 4, it depicts the strain energy per unit area when the wafers are fully bonded under different distributions. As the number of particles increases, the strain energy also increases. Meanwhile, the analytical results indicate that the Face distribution has the greatest impact on bonding, the Complex distribution has the least impact, and the differences among different distributions are relatively small.

Fig. 4

Strain energy per unit area of the wafer when fully bonded under different distributions

The Abaqus finite element method was used to simulate the wafer bonding process with different types of particle distributions. The parameter settings of the finite element method are shown in Table 1.

Table 1 The parameter settings of the finite element method

As shown in Fig. 5, in order to improve computational efficiency, a quarter—model was used for the simulation. To ensure that the upper wafer can be fully bonded under different distributions, a uniform pressure of 8000 Pa was applied in the normal direction of the upper wafer. Since the lower wafer serves as a fixed support, the lower wafer was set as a rigid body, and a simply—supported boundary condition was applied to the perimeter of the lower wafer. After meshing, 53690 elements were generated on the upper wafer.

Fig. 5: Finite element model diagram.

a Quarter-model. b Actual cross-section used for analysis²⁹

As shown in Fig. 6, they are the displacement diagrams and strain energy diagrams of the wafers under different distributions. Under pressure of 8000 Pa, the displacement at the edge of the wafer is close to the initial curvature of 0.1 mm, and it can be considered that the wafers are fully bonded.

Fig. 6: Wafer displacement and strain energy under different distributions.

(a) Cluster (b) Complex (c) Face (d) Line

As shown in Figs. 7–10, different particle distributions have different effects on the stress of the wafer. For the normal stress S_yy, the Complex distribution has the greatest impact, and the Line distribution has the least impact. For the shear stress S_yx, the Cluster distribution has the greatest impact, and the Line distribution has the least impact. For the shear stress S_yz, the Face distribution has the greatest impact, and the Line distribution has the least impact. Figure 11 shows the displacement after bonding under different distributions, the results indicate that different cases of particle distribution cause different displacement. Displacement at the edge of the wafer is greater than the internal displacement. The complex distribution displacement is greater than the remaining cases, but the difference is not significant.

Fig. 7: Plot of the stress after bonding under different distributions.

(a) Cluster (b) Complex (c) Face (d) Line

Fig. 8: Stress in the yy direction after bonding under different distributions.

(a) Cluster (b) Complex (c) Face (d) Line

Fig. 9: Stress in the yx direction after bonding under different distributions.

(a) Cluster (b) Complex (c) Face (d) Line

Fig. 10: Stress in the yz direction after bonding under different distributions.

(a) Cluster (b) Complex (c) Face (d) Line

Fig. 11: Plot of the displacement after bonding under different distributions.

(a) Cluster (b) Complex (c) Face (d) Line

As shown in Fig. 12, the simulation results indicate that the Face distribution has the most impact on bonding, which is consistent with the analytical results. However, the simulation results show that the Line distribution has the least impact on bonding, while the analytical results suggest that the Complex distribution has the least impact. The main reason for this difference is that during the bonding process, in addition to the main voids generated by the particles, a series of void tails are also produced³¹. These void tails have the same circumferential position θ_i. For the Line distribution, the void trails generated by the primary void all share the same circumferential position θ_i. Due to this alignment, void trails from different particles overlap to a certain extent, resulting in a smaller impact on bonding. In contrast, void trails produced under other distributions (Cluster, Complex and Face) do not overlap, correspondingly exerting a greater influence on bonding quality.

Fig. 12

Strain energy per unit area of the wafer when fully bonded under different distributions obtained from finite element calculation

As shown in Fig. 13, it is a comparison between the current work and the previous work²⁹. The previous work considered the influence of particles located at the same radial position r_i but different circumferential positions θ_i (i.e., the Face distribution in this paper) on bonding. For the other three distributions, since the radial positions r_i of the particles are different, in this work, the radial positions r_i of the particles are unified to r_imax (For the Cluster, Complex and Line distributions where particles have different radial positions r_i, this paper considers the effect of particle positions on bonding. While maintaining a constant particle count n, the radial positions of particles are unified to r_imax, with circumferential positions remaining unchanged. Additionally, due to the clustering of particles in the Cluster distribution, localized stress concentration may occur and affect bonding quality. The randomness of particle positions in the Complex distribution similarly impacts bonding performance, though this paper only considers one specific case of Complex distribution). The results show that the Cluster distribution is less affected by the radial position of the particles, while the Face and Line distributions are more significantly affected by the radial position of the particles.

Fig. 13: Comparison among analytical results, simulation results, and previous work²⁹ of strain energy per unit area when the wafer is fully bonded under different distributions.

(a) Cluster (b) Complex (c) Face (d) Line

Conclusion

Considering normal pressure and impurity particles, a mechanical model of wafer bonding with different particle distributions was established, and the analytical solution of the strain energy during the bonding process was derived. The specific conclusions are as follows:

• (1) The differences among different distributions are relatively small. Both the analytical and simulation results show that the Face distribution has the greatest impact on bonding. The linear particle distribution is more variable compared to the other cases. The reason is that during the bonding process, in addition to the main voids generated by the particles, a series of void tails are also produced.

• (2) The Cluster distribution is the least affected by the radial position of the particles, while the Face and Line distributions are more significantly affected by the radial position of the particles. This paper attributes the differences in bonding quality among distributions primarily to void trails. Additionally, potential stress concentration in Cluster distributions and the randomness of Complex distributions may also influence bonding performance. However, It should be acknowledged that this study only uses tungsten particles as impurities and considers a single shape function N(r, θ_i), leaving the consideration of impurities and shape functions incomplete.

• (3) This paper investigates the effects of different impurity particle distributions on wafer bonding and establishes corresponding mathematical models, which are validated through finite element simulation. The proposed approach provides theoretical guidance for wafer bonding processes in the presence of particles and offers reference value for other bonding forms.

• (4) In this paper, a wafer bonding model that takes into account different impurity distributions is formulated. The influence of impurities on bonding is manifested through the shape function N(r, θ_i). Although the numerical results and simulation results are in agreement, there is still a certain degree of error. Moreover. This paper does not explore the impact of different shape functions N(r, θ_i), and it remains unclear whether a more accurate shape function exists.

• (5) This paper addresses several impurity distributions that may occur in actual bonding processes. For Complex distributions, we randomly selected several coordinates (r_i, θ_i) as impurity positions without examining other configurations. However, the proposed model can compute bonding strain energy for particles at arbitrary positions within a certain range. According to the bonding criterion, wafer bonding feasibility can be determined by comparing the maximum strain energy with the adhesive energy. This provides practical guidance for wafer bonding processes.