Simulation of the bonding between silicon needle-like surfaces

The bond strength between two needle-like surfaces for the first bond attempt was simulated using experimentally obtained parameters, such as clustered needle density, height of clustered needles, diameter of clustered needles, distance between adjacent clustered needles, and bond interface width between two bonded needle-like substrates due to applied bonding loads (see Tables 3 and 4, and Figure 42). Required data for a given substrate (chip) size was then extracted and fed into the proposed bonding model programmed in the MATLAB (see Table 6).

129 For a given active bonding area (𝐴_𝑠), required number of needles (𝑁_𝑛) and gaps between needles (𝑁𝑔) were calculated using Eq. (66) and Eq. (67), respectively. Both maximum and minimum numbers of clustered needles and maximum and minimum values of bond interface widths were considered to take influences of their spreads into the account. Maximum values were obtained by addition of their mean values to their standard deviations (mean + STD), whereas the minimum values were obtained by subtraction of their standard deviations from their mean values (mean – STD). The total number of needles and gaps between them (𝑁𝑒) was then used to represent a needle-like substrate as a square binary matrix, where 1 represented a needle and 0 represented a 1 µm gap between needles. For this purpose, several needles and gaps had to be added to or subtracted from the total number of needles and gaps in order to get fit in a square matrix. The ratio between added or subtracted needles and gapes in each needle-like substrate was obtained according to the ratio between number of needles and gapes on that specific needle-like substrate. A normal distribution was then considered for distributions of 1s and 0s in the binary matrix. Afterwards, height (𝐻) and radius (𝑅) of needles were randomly generated with normal distributions considering their mean and standard deviation values and introduced to 1s elements of the binary matrix to represent geometry of needles.

Interaction of two square binary matrixes with different normal distributions were then used to represent two bonded chips. Deformations and interactions of needles were calculated according to the experimentally obtained bond interface widths due to the applied bonding loads for that specific needle-like surface (e.g., the needle-like surface 1). Both maximum (mean + STD) and minimum (mean - STD) bond interface widths were considered. The presented and displayed simulations results (see Figure 55 - 58) are the average of these four simulations (maximum number of needles and gaps with maximum bond interface widths, maximum number of needles and gaps with minimum bond interface widths, minimum number of needles and gaps with minimum bond interface widths, and minimum number of needles and gaps with maximum bond interface widths) and their corresponding spreads.

130

Table 6. Number of needles and gaps between needles and size of square matrixes used for simulations of the bond strength for the first bond attempt for the investigate needle-like

131 Due to the native oxidation process, a thin oxide layer (about 3 - 20 nm thickness [254]) covers surfaces of needles and substrates after their generation. Assuming needles as nano-porous needles [263], the oxide layer can make silicon dioxide as the interacting material between bonding pairs. Hence, silicon dioxide and its corresponding material parameters, such as yield strength (𝑌 = 8.4 𝐺𝑃𝑎 [264]), Young’s Modulus (𝐸 = 73 𝐺𝑃𝑎 [264]), surface energy (𝛾 = 35.82 × 10^{−3 𝐽}_𝑚² [265]), Poisson’s ratio (𝑣 = 0.17 [266]), and Hamaker constant (𝐻_𝐶 = 6.5 × 10⁻²⁰ 𝐽 [195]) were used to describe both needles and substrates materials. The breaking point (value) of a needle was calculated by Eq. (36) and Eq. (41). When the breaking point was reached, the corresponding needle was excluded from further considerations. For this purpose, a lateral displacement less than 25 % of diameter of the needle, and a maximum stress induced on the needle greater than 50 % of the silicon dioxide yield strength (assuming needles as nano-porous needles) were assumed as the breaking point of the needle while it was under a point load force, which acted on it with an angle of 75° with respect to x-axis at the beginning of the interlacing process. The applied bonding load was divided by number of needles to estimate the amount of the point force (𝐹_𝑛) acting on the needle. Vertical component of the point force (𝐹_𝑛𝑠𝑖𝑛 75 °) was considered as the normal point force (𝐹) acting on the needle, whereas its horizontal component (𝐹_𝑛𝑐𝑜𝑠 75°) was considered as the shear point force (𝐹_𝑠ℎ) acting on the needle. At a point where the needle started touching the opposite substrate, Eq. (32) was used to estimate compression length of the needle. At the end, the total bond strength (𝐹_𝑎𝑑ℎ) considering all contributing needles with respect to the applied bonding load was calculated using Eq. (42).

In the proposed VdW force model, the adhesion force between a needle and a substrate (𝐹_{𝑁−𝑉𝑑𝑊}) was obtained by Eq. (48), and Eq. (50) was used to obtain the adhesion force between two interlaced needles (𝐹𝑆−𝑉𝑑𝑊). To calculate these adhesion forces, the closest distance of approach (“0.3 nm” [32, 267, 268]) was assumed as the distance between the two interlaced needles (𝑑^′) and the needle and the substrate (𝑑) at their contact points.

In the proposed contact mechanics model, the distance between two interlaced needles (𝑑^′) and a needle and a substrate (𝑑) at their contact points was considered as zero. A Tabor parameter of 𝜇_𝑇= 0.1 [269–271] was assumed to obtain the adhesion

132

stress of a needle. Eq. (52) was then employed to calculate the adhesion force between the needle and the substrate (𝐹_{𝑁−𝐶𝑀}), and the adhesion between interlaced needles (𝐹𝑆−𝐶𝑀) was obtained by Eq. (54).

In the proposed capillary force model, the Kelvin length of water at 25°C (𝜆_𝐾 = 0.52 𝑛𝑚 [126]) was used as the reference value, and radius of curvature of a

meniscus for a given relative humidity (see Table 4) was calculated using Eq. (22).

The surface tension of water was considered as 72 × 10^{−3 𝐽}_𝑚² [228]. The same contact angle (𝜃₁ = 𝜃₂) between the enclosed water and involved surfaces was assumed for both needle-substrate and needle-needle interactions, and its corresponding value for a given filling angle (𝛽 = 0.2 °) was calculated using Eq. (59). For the case, where needles touched each other or substrates, a range of 0.3 - 0.7 nm was considered as the separation distance between two interacting surfaces (𝑑^′and 𝑑) in order to calculate the contact angles 𝜃₁ and 𝜃₂. The adhesion force between a needle and a substrate (𝐹𝑁−𝐶𝑎𝑝) was obtained by Eq. (62), and Eq. (65) was used to obtain the adhesion force between interlaced needles (𝐹𝑆−𝐶𝑎𝑝).

Figures 55 - 58 show simulation results of the bond strengths based on all proposed

adeshion modles (the VdW force, the contact mechanics, and the capillary force (with 𝑑 = 𝑑^′ = 0.7 nm)) in comparison with the measured bond strengths for the

bonded chips from the needle-like surfaces 1 - 4.

133 Figure 55. Simulated and measured bond strengths of the bonded chips from the needle-like surface 1. Active bonding area: 0.25 cm².

Figure 56. Simulated and measured bond strengths of the bonded chips from the needle-like surface 2. Active bonding area: 0.36 cm².

0 200 400 600 800 1000 1200

0 2 4 6 8 10 12 14

Bond strength (kPa)

Applied bonding load (kg)

Van der Waals Contact mechanics (DMT)

Capillary (ß=0.2°) Measurement

0 30 60 90 120 150 180 210 240

0 2 4 6 8 10 12 14

Bond strength (kPa)

Applied bonding load (kg)

Van der Waals Contact mechanics (DMT)

Capillary (ß=0.2°) Measurement

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Figure 57. Simulated and measured bond strengths of the bonded chips from the needle-like surface 3. Active bonding area: 0.36 cm².

Figure 58. Simulated and measured bond strengths of the bonded chips from the needle-like surface 4. Active bonding area: 0.25 cm².

0 20 40 60 80 100 120

0 2 4 6 8 10 12 14

Bond strength (kPa)

Applied bonding load (kg)

Van der Waals Contact mechanics (DMT) Capillary (ß=0.2°) Measurement

0 10 20 30 40 50 60

0 2 4 6 8 10 12 14

Bond strength (kPa)

Applied bonding load (kg)

Van der Waals Contact mechanics (DMT)

Capillary (ß=0.2°) Measurement

135