疊晶構裝是指將一個以上的裸晶，以高度方向堆疊封裝在同一個構裝體中。其目的在於節省空間、提高性能速度、降低功率消耗、節省封裝，乃至於整個封裝的成本。然因晶片相互堆疊，使各材料間偶合行為趨於複雜，疊晶構裝可靠度問題儼然已成為亟待解決的課題。
在探討疊晶構裝錫球可靠度時，因疊晶構裝體元件組成複雜，一般除實驗測試外，有限元素分析軟體為常用的數值模擬方法。對於構裝體的錫球配置型態無法採更簡化的模型進行分析者，若以整體模型分析法直接求解，除龐大的單元數及節點數會導致大量計算時間外，由於錫球以黏塑性本構性質模擬，為精確分析錫球行為，勢必將網格進行很精細的分割，因而耗費巨大的計算時間、成本與龐大的資料儲存空間，甚至不可行。
為解決上述問題，本文基於全域/局部分析法結合最佳化演算法，發展一簡單、有效率能預測構裝體錫球的變形現象及疲勞壽命的方法－最佳等效錫球觀念修正子模型分析法，其特點為可以替代全域/局部分析方法，並驗證立方等效體積錫球子模型法所產生的誤差大的結果。本法因引入最佳等效錫球於整體模型中，可將模型的單元數及節點數急遽減少，不但可兼顧計算精度及計算效能，尤其在使用實驗計劃法進行最佳化參數分析時，可以大幅縮短分析所需的時間。
為驗證所發展的最佳等效錫球觀念修正子模型分析方法之精確性與效率，實際驗證於WB-SC疊晶構裝體分析模型中。構裝體組成元件包含熱黏膠、真實晶片/間隔晶片、基板、印刷電路板及錫球(Sn63/Pb37)，錫球採用亞蘭德黏塑性本構模式考慮錫球的黏塑性行為，分析錫球應變能密度代入Darveaux能量疲勞模型進行疲勞可靠度分析。與整體精細網格模型分析法、全域/局部分析法、立方等效體積子模型分析法進行比較，發現使用本法僅須整體精細網格模型18.8%的計算時間，且所需的硬碟容量也只要整體精細網格模型的9.98% ；另本法與整體精細網格模型的分析結果在溫度循環負載下其循環間累積應變能密度差異僅有5.76%且趨勢一致；再者，本法與全域/局部模型結果一致，足見本法可替代全域/局部分析方法，故本文所發展的最佳等效錫球觀念修正子模型分析法進行有限元素分析，一方面可以節省分析時間及兼顧其分析的精確性，另一方面又可減少分析結果檔所需的儲存資源。
其次，針對WB-SC BGA疊晶構裝體進行單因子分析，幾何參數包括真實晶片厚度、間隔晶片厚度、基板厚度、PCB厚度；材料參數包括封膠楊氏模數、基板楊氏模數、封膠熱膨脹係數及基板熱膨脹係數等共八個參數，全面性探討各參數對構裝體應變能密度及錫球疲勞壽命的影響。隨後將以錫球間應變能密度分佈觀點進一步探討疊晶構裝體疲勞壽命，乃提出一整合目標D值，即將描述應變能密度均勻分配的指標davg及錫球間應變能密度之最大與最小差異指標ddiff整合為單一目標，以便於實驗計劃法的分析。最後，以反應曲面結合基因演算法進行參數最佳化設計，及探討各參數間的交互作用對整合目標D值的影響。
首先，單因子分析結果顯示，較厚的真實晶片與間隔晶片、較厚的基板、較薄的電路板、楊氏模數較低的封膠、楊氏模數較高的基板、熱膨脹係數較低的封膠以及熱膨脹係數較高的基板，皆能有效提高疊晶構裝體疲勞壽命。
接著以反應曲面法分析，然為降低同時考量幾何與材料因子所需之反應曲面法實驗次數，提出：(1)不考慮幾何與材料因子間的交互作用，分別建立個別的反應曲面之雙反應曲面法。(2)考量幾何與材料因子間交互作用的存在，並配合部分因子設計法篩選設計參數，建立單一偶合反應曲面之混合反應曲面法，再將所得反應曲面模型採用基因演算法(Genetic algorithm)進行最佳參數組合搜尋，以得到最佳疲勞壽命。至於雙反應曲面法所建構的幾何與材料的兩反應曲面皆經統計檢定，以確認模型的配適性，而最終所得二反應曲面皆為一修正的二階配適模型(Reduced Quadratic Model)；而混合反應曲面模型經統計檢定，確認模型的配適性後，則為一修正的立方配適模型。
由雙反應曲面法的幾何反應曲面結合基因演算法所預測的整合目標D值為0.2257 MPa(或MJ/m3)，與實際實驗所得到的整合目標D值為0.2255 MPa相近，可知幾何反應曲面所預測的值相當準確，同時與單因子分析法所得到的最佳製程參數組合恰相同；在材料參數最佳化設計方面，所預測的整合目標D值為0.1465 MPa，與實際實驗所得到的整合目標D值為0.1665 MPa相近，亦驗證材料反應曲面相當準確；最後再合併修改幾何與材料參數水準至最佳直，經實際實驗所得到的整合目標D值為0.1267MPa，而對應的錫球疲勞壽命為1002個循環，故與單因子分析法所得到的結果804個循環更佳，較原始構裝體壽命大幅提升123.7%，至於整合目標D值也由原始構裝體的0.3123 MPa與單因子分析法的0.1464 MPa，下降至0.1267 MPa。另將反應曲面法所求得各控制因子對整合目標D值的影響趨勢與單因子分析得到的結果進行比較，其趨勢一致，亦進一步驗證所建立的雙反應曲面之可靠性。
接著混合反應曲面模型結合基因演算法所得之錫球疲勞壽命為1014個循環，與雙反應曲面法所得到的結果1002個循環更佳，較原始構裝體提升126.3%，而整合目標D值也由原始構裝體的0.3123 MPa、單因子分析法的0.1464 MPa與雙反應曲面法的0.1267 MPa，下降至0.1258MPa，因此最佳化過程確實有效。此外，由變異數分析表知，雖混合反應曲面法所得之davg值0.0827MPa較雙反應曲面法所得之davg值0.0824MPa為差，然混合反應曲面法所得之ddiff值0.1912MPa較雙反應曲面法所得之ddiff值0.1914MPa為佳，可見由單一個davg指標或ddiff指標並無法作為疊晶構裝體疲勞壽命最佳化之目標函數，而整合目標D值與疊晶構裝體疲勞壽命一致，由此更驗證其有效性。又透過混合反應曲面法與雙反應曲面法各參數間的反應曲面與等高線圖作比較，可發現大致上趨勢相符，亦進一步驗證所建立的混合反應曲面之可靠性。
總之，透過本論文所發展之分析方法、錫球間應變能密度分佈評估指標、基因演算法結合雙反應曲面與混合反應曲面，可快速評估疊晶構裝體錫球累積應變能密度受設計參數的影響，以供構裝設計時之參考。

A stacked die package consists of a plurality of chips stacked up one over another for the purposes of reducing occupied space, enhancing functional speed, lowering power dissipation, miniaturizing the structure, and consequently cutting down the costs of the entire package. Due to the stacked structure of the package, the coupling behavior between materials becomes complicate so that the reliability of stacked die package becomes a critical issue to be solved.

To study the reliability of solder balls in a stacked die package with complex components, the finite element analysis software is always adopted in addition to the experimental tests. Once a simplified model to the solder ball array in a package can not be applied in the software analysis, the application of a global model to the analysis would cause enormous computing time due to huge amount of elements and nodes. Meanwhile, and a constitutive model which simulates the viscoplasticity of solder balls requires very fine meshing for the purpose of accurate analysis so that immense computing time, costs and data storage space would make it an impossible job.

In this paper, a global/local method along with an optimization algorithm, so called the modified submodeling approach of optimal equivalent solder, is developed as a simple but effective approach to predict the deformation and the reliability of solder balls in the package. Such a method can replace the global/local method and verify that the cubic equivalent volume sub-modeling method leads to bigger errors and inaccurate displacement field. The comprisal of equivalent solder in this method is facilitated to obviously reduce the number of elements/nodes so as to not only enhance computing accuracy and efficiency but also save computing time in the analysis of optimal parameters by the design of experiment.

To verify the accuracy and efficiency for the modified submodeling approach of optimal equivalent solder, a model of WB-SC stacked die package comprised of die adhesive, real chips/spacer chip, substrate, printed circuit board and solder balls (Sn63/Pb37) is simulated and analyzed. The Anand viscoplasticity constitutive model is adopted to the solder alloys. For the analysis of the strain energy density of solder balls, the Darveaux energy-based fatigue model is taken into to analyze the fatigue reliability of solder balls. Compared with the global detailed mesh method, the global/local method and the cubic equivalent volume sub-modeling method, it is found that the computing time and the hardware space required of the method adopted in this paper are only 18.8% and 9.98% of that by the global refined mesh model respectively. Moreover, under thermal cycling load, the accumulated strain energy density difference between the two models is merely 5.76 %, and the trends of two models coincide with each other. On the other hand, the adopted method is eligible to replace the global/local method since the results of two methods are completely accordant with each other. Therefore, the modified submodeling approach of optimal equivalent solder developed in this paper is recognized as an effective method in computing time and data storage space saving as well as in the analytical accuracy increasing.

Additionally, one-factor-at-a-time analysis to the WB-SC BGA stacked die package is conducted in which geometric parameters such as real chip thickness, spacer chip thickness, substrate thickness, PCB thickness and material parameters such as Young’s modulus of encapslant, Young’s modulus of substrate, CTE of encapslant and CTE of substrate are considered. The effects of these 8 parameters on the strain energy density of package and the fatigue life of solder balls are investigated. Subsequently, from the viewpoint of strain energy density distribution for solder balls, the fatigue life of stacked die package is investigated. Then an integrated objective value D is proposed, which integrates the uniform distribution index of strain energy density davg, and the index of the difference between maximum strain energy density and minimum strain energy density, ddiff, into a single objective for the analysis of the experiment design method. Finally the response surface is combined with the genetic algorithm to work on the optimal design of parameters as well as analyze the effect of the interaction among parameters on the integrated objective value D.

First of all, the single factor analysis indicates that thicker real chip/space chip, thicker substrate, thinner PCB, encapslant with smaller Young’s modulus, substrate with larger Young’s modulus, encapslant with smaller CTE and substrate with larger CTE, are always facilitated to increase the fatigue life of stacked die package.

Next, the response surface method is adopted. To reduce the experiment frequency while geometric factors and material factors are both involved, the following actions are proposed: (1) Neglect the interaction between geometric factors and material factors and set up two individual response surfaces so-called the double response surface method. (2) Take the interaction between geometric factors and material factors into account, and apply the fractional factorial design to select design parameters; then set up a coupled response surface so-called the mixed response surface in which genetic algorithm is introduced to search for the optimal combination of parameters to obtain the optimal fatigue life of solder balls. Furthermore, both the geometric and material response surfaces from the double response surface method are affirmed to be compatible through statistics verification, and the response surface either is a reduced quadratic model, whereas the mixed response surface is recognized as a reduced cubic model.

By combining the geometric response surface in the double response surface method with the genetic algorithm, the integrated objective value D is estimated to be 0.2257 MPa(or MJ/m3), which is very closed to the real experimental value D, 0.2255 MPa. This indicates that the prediction by the geometric response surface method is quite accurate. Moreover, both the double response surface method and the one-factor-at-a-time techniques result in the same optimal combination of parameters. As for the optimal design of material parameters, the integrated objective value D is estimated to be 0.1465 MPa, which is quite closed to the real experimental value D, 0.1665 MPa. This again ensures that the material response surface performs very well. Next, the geometric parameter level and the material parameter level are combined together and revised to reach an optimal value. Through the real experiment, the integrated objective value D is obtained as 0.1267MPa, which is less than that of the original package, 0.3123 MPa and the one-factor-at-a-time techniques, 0.1464 MPa. Nevertheless, the fatigue life of solder ball is 1002 cycles, which is better than 804 cycles from the one-factor-at-a-time techniques, and 123.7% greater than that of the original package. On the other hand, the effects of control factors on the integrated objective value D for the response surface method and the one-factor-at-a-time techniques are compared with each other and found to be coincident in trend. This once again verifies the reliability of the double response surface method.

Accordingly, by combining the mixed response surface method with the genetic algorithm, the fatigue life of solder balls is obtained as 1041 cycles which is better than 1002 cycles from the double response surface method and 126.3% greater than that of the original package. Yet the integrated objective value D from the mixed response surface method is obtained as 0.1258 MPa which is less than 0.3123 MPa from the original package 0.1464 MPa from the one-factor-at-a-time techniques and 0.1267 MPa from the double response surface method. This indicates that the optimization of the mixed response surface method works well.

Furthermore, the analysis-of-variance table shows that although the davg value of 0.0827 MPa from the mixed response surface method is greater than 0.0824 MPa from the double response surface method, thus the ddiff value of 0.1912MPa from the former method is less than 0.1914MPa from the later method. This declares that neither index davg nor index ddiff can be the optimal objective function, yet only the integrated objective value D, which is accordant with the fatigue life of the stacked die package, is proved to be more effective. In addition, the response surfaces of parameters for both the mixed response surface method and the double response surface method are consistent with the contour plots so that the mixed response surface method is again proved to be reliable.

In conclusion, the analysis methods, the evaluation index of strain energy density between solder balls as well as the genetic algorithm combined with the mixed response surface method/double response surface method proposed and developed in this paper is eligible to efficiently predict the effect of the design parameters on the accumulated strain energy density of solder balls in stacked die package so that it indeed provides valuable advices for the package design.

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