雷射鑽孔技術挾其高精度、高加工品質之優勢，在矽通孔(Through-Silicon-Via, TSV)領域上逐漸嶄露頭角；其中又以飛秒雷射對材料周遭結構破壞小、熱影響區域小，使其最具潛力。然而影響通孔品質之研究卻尚止於逐因的定性比較，因此本研究欲以實驗設計方法，通盤考量影響通孔品質之可能因素，探討其顯著性與作用關係，再建立因子對通孔品質影響的回歸模型，並找出最佳因子組合以提升通孔品質或降低能耗成本。本研究成功以分子動力學法模擬飛秒雷射用以加工矽通孔，達到減少實驗成本之目的，並採用真圓度作為量化飛秒雷射矽通孔品質之指標，並細分為入口、出口與孔內真圓度。
透過兩階段實驗設計法規畫實驗，先以部分因子設計篩選出對通孔真圓度具顯著影響的因子，再用完全因子設計分析顯著因子之主要與交互作用，及其貢獻度；並以正向分析Yates運算法建立顯著因子對各孔洞真圓度影響的回歸模型。而計算得回歸模型之配適度均在95%以上，說明各回歸模型皆能高度反映實驗結果。
另外，本研究透過調控貢獻度，以逆向Yates運算對孔洞真圓度進行優化，考慮到業界多元的設計需求，提供兩種優化策略：(1) 依給定真圓度求最低能耗；(2)依指定能耗作最佳真圓度。成功地在指定設計需求下，求出各個真圓度在兩種優化策略中的最佳因子組合。

SUMMERY

In this study, experimental design method are used to analyze the significance and interaction of possible system factors to femtosecond laser drilled hole quality in Through-Silicon-Via (TSV). Circularity of hole is used to characterize the hole quality and is simulated by molecular dynamic simulation to reduce experimental cost. Circularity is divided into three portions including entrance, exit and in-hole circularity. Using a two-step experimental design approach, significant factors that have potential influence on hole circularity are selected via fractional factorial design. Exhaustive factorial design is then used to analyze the major and interaction effects among significant factors. Followed by that, the corresponding mathematical model is thus established via Yates analysis algorithm. The goodness of fit of three circularity models are computed and the results presenting the well fit of the model higher than 95% of the experiment results are achieved. In addition, inverse Yates method is also used to optimally design the hole circularity by readjusting the factorial contribution. To serve for various design objective, two optimization strategies are employed including (1) optimization for minimal energy cost with given circularity, and (2) optimization for best circularity with given energy cost. The results are satisfactory and can be used for innovative design project for various laser drilling processes and can be used for industrial design application.

Key words: Femtosecond Laser, Through-Silicon-Via, Molecular Dynamics Simulation, Circularity, Experimental Design.

INTRODUCTION

With high accuracy and high manufacturing quality, laser drilling has become a popular technology in Through-Silicon-Via process. Since femtosecond laser technology possesses lower damage to surrounding material structure and small heat affected zone, gives it more potential for possible higher quality.

A number of experimental investigations have been performed by researchers such as Baudach, Wang and Yu in order to obtain the effect of different laser parameters on laser drilling results. It turns out that hole quality is affect by various laser parameters. However, there’s still has little studies on the effect of interactions between manufacturing factors. On the other hand, Herrmann, Watanabe, Liu, et al. used molecular dynamics simulation to study ultrafast laser interacts with material. It is shown that molecular dynamics simulation can be an effective tool for research of femtosecond laser drilling.

In this study, circularity is used to characterize laser drilled hole quality on silicon and is simulated by molecular dynamics simulation. Experimental design methods are employed to find out significant factors, to construct mathematical model and to optimize system by improving circularity or decreasing energy cost. The result is verified by simulation using optimized combination of manufacturing factors.

MATERIALS AND METHODS

Models of two different size, 30 nm × 30 nm × 100 nm and 30 nm × 30 nm × 200 nm, as well as two different crystal orientations, (100) and (110), are established for molecular dynamics simulation. Pulse width of laser is 200fs. Tersoff potential is employed to calculate interacting force of atoms. Position and velocity are integrated by velocity Verlet algorithm. Laser energy is transformed into molecular kinetic energy by rescaling atoms’ velocity. The circularity is divided into three portions according to its position including entrance, exit and in-hole circularity.

Considering laser parameter, silicon material and geometrical effect, seven factors are selected in this study. Seven factors are x1 pulse energy, x2 pulse frequency, x3 machining time, x4 material thickness, x5 lattice direction, x6 hole diameter and x7 focus position. Fractional factorial design is used to select significant factors that have potential influence to hole circularity. Exhaustive factorial design is used to analyze significance and contribution of major and interaction effects. Yates analysis algorithm is then used to establish system mathematical model.

Two strategies to optimize laser drilling process are proposed in this study including (1) optimization for minimal energy cost with given circularity, and (2) optimization for best circularity with given energy cost. The inverse Yates method is employed to proceed optimization by readjusting factors’ contributions. To prevent missing optimal solution, factors’ contributions are readjusted through three ways including (a) enhance high contribution (above 15%), (b) enhance middle contribution (10%~15%) and (c) reduce low contribution (below 10%).

RESULTS AND DISCUSSION

Entrance Circularity
The significant factors analyzed are x1 pulse energy, x2 pulse frequency, x5 lattice direction and x6 hole diameter. The provincial mathematical model of system is

and the goodness of fit is 96.21%. The optimization for minimal energy cost with given circularity reduce energy cost form 50 mJ to 42.93 mJ. And the optimization for best circularity with given energy cost improves circularity form 2.1 nm to 1.66 nm.

Exit Circularity
The significant factors analyzed are x1 pulse energy, x2 pulse frequency, x5 lattice direction and x6 hole diameter. The provincial mathematical model of system is

and the goodness of fit is 95.39%. The optimization for minimal energy cost with given circularity reduce energy cost form 50 mJ to 44.8 mJ. And the optimization for best circularity with given energy cost improves circularity form 2.21 nm to 1.86 nm.

In-hole Circularity
The significant factors analyzed are x1 pulse energy, x2 pulse frequency, x4 material thickness and x6 hole diameter. The provincial mathematical model of system is

and the goodness of fit is 95.23%. The optimization for minimal energy cost with given circularity reduce energy cost form 50 mJ to 46.25 mJ. And the optimization for best circularity with given energy cost improves circularity form 2.16 nm to 1.91 nm.

CONCLUSION

Molecular dynamics is used for simulating femtosecond laser drilling on silicon. Entrance, exit and in-hole circularity are obtained from simulations of seven different manufacturing factors. Experimental design methods are employed to select significant factors for each circularity, to establish mathematical model of system and to optimize laser drilling process under different strategies.

The simulation result in this study is more ideal than experiment. However the error would be less than 5%, the result should be practical. Considering the interaction of significant factors, the mathematical model of system would be more accuracy and improvement would be more reasonable.

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