Process-Quality Evaluation for Wire Bonding With Multiple Gold Wires

Taiwan is a world-leader in wafer foundry services and IC packaging and testing. Wire bonding is a crucial process in the overall IC-packaging industry chain. Thus, this paper proposes a process-quality evaluation model for wire bonding with multiple gold wires. We chose process quality indices as a tool of evaluation fully mirroring process yield and quality levels. These indices contain unknown parameters and thus require sample data to estimate. We first derived the uniformly minimum variance unbiased estimator of the indices and calculated the upper confidence limits of the indices based on DeMorgan’s theorem and Boole’s inequality. The upper confidence limits of the indices were then employed to create a confidence interval-based fuzzy membership function, in order to improve the accuracy of estimation as well as solve the problem of uncertainty of the measured data. Next, we obtained the fuzzy critical value and used index estimates and the fuzzy critical value to establish fuzzy test rules. Next, we marked the fuzzy critical value on the axes of a radar chart, which is a visualization evaluation tool, and connected neighboring critical points to create a critical region in the form of a regular polygon. The observed values of the indices were then marked on the axes to produce a visualized fuzzy radar evaluation chart. This fuzzy radar evaluation chart has a solid foundation in statistical inference, and evaluation rules were established using precise fuzzy test methods. Not only is this fuzzy radar evaluation chart easy to use, but it also reduces the chance of misinterpretations made by sampling errors, so that the accuracy of evaluation can be enhanced.


I. INTRODUCTION
Taiwan's wafer foundry output accounts for approximately 70% of the global market, and its IC packaging output occupies roughly 50% of the global market, marking Taiwan out as a world-leader in the industry. Chen et al. [1], Tseng et al. [2], and Tunn et al. [3] suggested that the cluster effect of Taiwan's electronics industry has resulted in an industrial eco-chain for information-and-communication technology (ICT), which dominates the electronics industry worldwide. In addition, Taiwan has a complete industry chain from IC design, foundry services, and packaging and testing all the way to production and assembly [4]. IC packaging includes wafer dicing, die bonding, wire bonding, The associate editor coordinating the review of this manuscript and approving it for publication was F. K. Wang . encapsulation, packing, singulation and lead forming, marking, and lead finishing, as shown in Figure 1. In this study, we focus on wire bonding, which involves welding gold wires onto the inner leads of chips and lead frames. These wires are vital media for electrical connections and signal transmissions between the internal and external circuits of ICs [5], [6].
We therefore propose a fuzzy process-quality evaluation model for wire bonding in IC packaging. Fuzzy evaluation models not only assess process quality but also identify which critical-to-quality (CTQ) characteristics require improvement [7]- [10].
Process capability indices (PCIs) are tools for processquality evaluation commonly applied in the industry. They can help manufacturers evaluate the process quality of their production processes and serve as an effective and convenient means of communication between internal engineers [11]- [13]. Six Sigma, developed in 1986 by Motorola, is widely applied to enhance product quality levels in the manufacturing industry [5], [14]- [16]. Several studies have investigated the correlations between PCIs and Six Sigma quality levels [9], [10], [17]. The wire-bonding process has two significant quality characteristics, both of which are the larger-the-better (LTB) type. Based on the work of Chang et al. [18], this study proposes the following processquality index suitable for assessing LTB quality characteristics. According to Aslam [19], there is a one-to-one mathematical relationship between this index and process yield presenting Six Sigma quality levels.
Obviously, process quality index can completely reflect the process yield and quality level. Therefore, the proposed index is employed to evaluate the quality of the wire-bonding process for IC packaging. This process-quality index involves unidentified parameters, so sample data is needed for estimation [20]- [22]. Many researchers have created fuzzy evaluation models [15]- [23] using the confidence intervals of indices, attempting to improve the accuracy of estimation and overcome uncertainty in the measured data. We first derived the upper confidence limit of the proposed index and employed the method used by Buckley [26] and Chen and Chang [27] to create a confidence-interval-based fuzzy membership function. Then, we obtained the fuzzy critical value and used index estimates and the fuzzy critical value to establish fuzzy test rules. Finally, an easy-to-use visualized radar chart serves as the evaluation interface. The radar chart is a visualization tool widely used in fields such as engineering, management, and education [28]- [31]. Not only is this fuzzy radar evaluation chart easy to use, but it also diminishes the chance of misinterpretation due to sampling errors, thereby increasing the accuracy of evaluation.
The rest of this paper is arranged as follows. Section 2 demonstrates the estimations of process-quality indices and finds a uniformly minimum variance unbiased estimator (UMVUE) of index PQIL and its 100 (1 − α) % upper confidence limits. Section 3 applies these upper confidence limits to derive the fuzzy critical value. Next, index estimates and the fuzzy critical value are used to establish fuzzy test rules. Based on these rules, Section 4 constructs an easyto-use visualized radar evaluation chart to serve as an evaluation interface and an evaluation procedure is established. Section 5 presents our conclusions.

II. ESTIMATIONS OF PROCESS-QUALITY INDICES
As previously mentioned, IC and ceramic packages generally have several gold wires, as shown in Figure 2. In an attempt to prevent loss of generality, we assume that an IC packaging process involves l gold wires, each of which has two LTB quality characteristics: will pull strength (h = 1) and ball shear strength (h = 2). Let random variable X jh denote the process distributions of quality characteristic h for gold wire j. Then X jh is distributed normally with mean µ jh and standard deviation σ jh , where h = 1, 2 and j = 1, . . . , l. The process quality indices can be defined as follows: where LSL jh stands for the lower specification limit. Based on Chang et al. [18], we know that µ jh − kσ jh = LSL jh indicates that the process quality attains the k − sigma level, and therefore, Clearly, when PQIL jh equals k, the process quality exceeds the k − sigma level. Furthermore, there is a one-to-one mathematical relationship between index PQIL jh and process yield p: where Z = X jh − µ jh σ jh follows standard normal distribution and (z) is the cumulative distribution function for standard normal distribution. Based on Chang et al. [18], we let X jh1 , · · · , X jhi , · · · , X jhn be a random sample of X jh with sample size n. The estimators of µ jh and σ jh are expressed as follows: and Therefore, the estimators of the process-quality indices are as follows: where b n is the correction factor expressed as follows: Obviously, if n approaches infinity, then b n = 1. Under the assumption of normality,X jh − LSL jh is distributed as N µ jh − LSL jh , σ 2 jh n and S −1 jh is distributed as Obviously, the expected value ofX jh − LSL jh is µ jh − LSL jh and the expected value of S −1 jh can be expressed as follows: SinceX jh and S 2 jh are mutually independent, the expected value of PQIL * jh can be expressed as follows: Therefore, the unbiased estimator PQIL * jh is only a function of X jh , S 2 jh . Therefore, PQIL * jh is the UMVUE ofPQIL * jh . Obviously, the distribution of b −1 n × √ n × PQIL * jh is a non-central t-distribution with n − 1 degrees of freedom and the non-centrality parameter δ is √ n × PQIL * jh , denoted as t n−1 (δ).
Let the random variable Z be as follows: Then random variable Z follows standard normal distribution. Therefore, and (12) where Z α/2 is the upper α 2 quantile of N (0, 1) and χ 2 1−α/2;n−1 is the lower 1 − α 2 quantile of χ 2 n−1 . Furthermore, let event E jh1 and event E jh2 be as follows: Then the complement of event E jh1 and event E jh2 can be shown as follows: Based on DeMorgan's rule and Boole's inequality, Thus, Equivalently, Suppose the observed values of X jh1 , · · · , X jhi , · · · , X jhn are x jh1 , · · · , x jhi , · · · , x jhn . Then,x jh and s jh are respectively the observed values ofx jh and s jh as shown below: x jhi (16) and Thus, the observed value of upper confidence limit for index PQIL jh is where PQIL * * jh0 is the observed value of PQIL * jh × b −1 n as displayed below.

III. FUZZY HYPOTHESIS TESTING
Statistical hypothesis testing is an effective method by which to determine if the process-quality level is legitimate. As stated above, when PQIL jh = k, this indicates that the process quality has reached the k − sigma level. Hypothesis testing at the significance level α is expressed as follows: As mentioned before, the null hypothesis H 0 is PQIL jh ≥ k and the alternative hypothesis H a is PQIL jh < k.
Therefore, the statistical hypothesis testing rules are as follows: (1) If PQIL * * jh0 ≥ k S , then do not reject H 0 and conclude that process quality meets requirement.
(2) If PQIL * * jh0 < k S , then reject H 0 and conclude that process quality does not meet requirement. Using the abovementioned rules and the methodology proposed by Chen [20], we developed a fuzzy testing method based on the observed value of the upper confidence limit for index PQIL jh . As described by Chen [20], the α-cuts of the triangular fuzzy number PQIL * * jh0 are as follows: The half-triangular fuzzy number of PQIL jh is PQIL * * jh0 = PQ jhM , PQ jhR , where Therefore, the membership function of the fuzzy number PQIL * * jh0 is (26) where α is determined by PQIL * * jh0 (α) = x. Figure 3 exhibits membership function η jh (x) with vertical line x = k.
Based on Chen et al. [23] and Chen [20], we let set A Tjh be the area of η jh (x) and set A Rjh be the area of η jh (x) to the right of the vertical line x = k. Then, and   Obviously, when PQ jhM = k, d Rjh 2d Tjh = 1 2. On the basis of the approach adopted by Chen et al. [23] and Buckley [26], we let 0 < φ ≤ 0.5 and let the decision value dv jh of quality characteristic h of gold wire j satisfy the following equation: Then decision value dv jh = PQ jhR − 2φd Tjh can be shown as follows: Based on Chen et al. [23], the decision rule for fuzzy testing is (1) If k ≥ dv jh is equivalent to d Rjh 2d Tjh ≤ φ, then reject H 0 and conclude that PQIL jh < k. (2) If k < dv jh is equivalent to d Rjh 2d Tjh > φ, then do not reject H 0 and conclude that PQIL jh ≥ k.
We set k F as a fuzzy critical value. Figure 4 displays the relationship between sample size n and the critical values. k S < k F < k can be distinguished clearly in Figure 4. Obviously, under the situation of the same sample size, k F will be closer to k value than k S . Therefore, fuzzy testing is more precise than statistical testing. Besides, whether k F or k S is a function of sample size n, and lim n→∞ k S = lim n→∞ k F = k. Then, the decision rules for the fuzzy testing are as follows: (1) If PQIL * * jh0 ≥ k F , then do not reject H 0 and conclude that PQIL * * jh0 ≥ k.

IV. EVALUATION PROCESS USING FUZZY RADAR EVALUATION MODEL
As stated above, in order to determine whether the process quality reaches the 5-sigma level, hypothesis testing at significance level.005 is as follows: H 0 : PQIL jh ≥ 5(meets the requirement) H a : PQIL jh < 5(does not meet the requirement) The fuzzy critical value k F calculated using Eq. (34) is then marked on the axes and connected to form a polygon radar evaluation chart. Using Eq. (19), 2 × l index values are then derived and marked on the 2×l axes to form a visualized evaluation chart. The example employed in this study involved an IC package with 6 gold wires (l = 6), which means that it has 12 quality characteristics. Table 1 displays the relevant data. Next, we explain this example using numerical methods to show how to apply the theory proposed in this study. Suppose that a company takes 36 random samples for testing. As the IC package has 6 gold wires (l = 6), and each gold wire has two quality characteristics (h = 2), then there are a total of 12 sets of sample data. To help manufacturers utilize the proposed evaluation model, we established the following evaluation procedure: Step1: Calculate the mean of samples, the standard deviation of the samples, and index estimates of the 12 samples using Eqs. (16), (17), and (19).
Step2: Based on Eq. (34), with n = 60 and φ = 0.3, the fuzzy critical value can be obtained as follows: Step3: Draw 12 axes at 30 degrees from each other, mark the fuzzy critical value (k F ) on the axes of the radar chart, VOLUME 8, 2020  and connect the neighboring critical points to form a critical region in the form of a regular dodecagon, as shown below: Step4: Mark the 12 index estimates PQIL * * jh0 (j = 1, . . . , l and h = 1, 2) in Table 2 on the 12 axes. The two ends of axis j respectively indicate the index estimates PQIL * * j10 and PQIL * * j20 of the two quality characteristics of gold wire j, as shown in Figure 6.
Step5: If index estimate PQIL * * jh0 falls within the twelve-sided critical region and PQIL * * jh0 ≤ 4.455, then reject H 0 and conclude that PQIL jh < 5. This means that this quality characteristic is in need of improvement. If index estimate PQIL * * jh0 falls outside of the twelve-sided critical region and PQIL * * jh0 > 4.455, then do not reject H 0 and conclude that PQIL jh ≥ 5.
Based on the test rules in Step 5, the quality-level index estimates PQIL * * jh0 for the two quality characteristics of wire 2 both fell within the critical region, which means that they did not meet the five-sigma quality requirement and are Based on statistical principles, these two upper confidence limits are greater than 5; therefore, we do not reject H 0 and conclude that PQIL jh ≥ 5. However, PQIL * * 210 = 4.201 and PQIL * * 220 = 4.422. Both values are obviously smaller than 5. That means the proposed model returns more reasonable results than the previous statistical test results, and is consistent with the discussions made by several studies [20], [24], [33]- [35].

V. CONCLUSIONS
This study proposes a fuzzy quality evaluation model for the wire-bonding process in IC packaging. Both of the quality characteristics of the gold wires are LTB quality characteristics, and for this reason, we propose a process-quality index suitable for LTB quality characteristics based on the concepts presented by Chang et al. [18]. This index fully exhibits quality levels and has a one-to-one mathematical relationship with process yield. It is thus a good assessment tool. Since this index involves unknown parameters, sample data are used to find the UMVUE of process-quality indices. Based on DeMorgan's rule and Boole's inequality, we developed the 100 (1 − α) % upper confidence limits of the indices. Subsequently, we elicited the fuzzy membership function on the basis of these upper confidence limits and obtained the fuzzy critical value to serve as a fuzzy evaluation standard, with which we developed a visualized fuzzy radar evaluation chart. This fuzzy radar evaluation chart has a solid foundation in statistical inference, and evaluation rules were established using precise fuzzy test methods. In addition, this chart constructs a whole picture concerning the important quality characteristics of the product. It facilitates management and aids suppliers in progressing and forming long-term relationships with their partners. Applied in practice, this model will not only benefit industry chains but will also improve the quality of their processes.