## 4 Design of Practical PADT Plan with Two Stress Variables
### First Phase: PADT Plan with Two Stress Levels by Each Variable
\begin{array}{c}\pi_{22}\pi_{31}\pi_{33}\eta_{22}^{2}\eta_{31}^{2} \eta_{33}^{2}(T_{2}-1)^{2}V_{2}^{2}\\ +\pi_{13}\eta_{13}^{2}\Bigg{(}\begin{array}{c}\pi_{22}\pi_{31}\eta_{22}^{2} \eta_{31}^{2}T_{2}^{2}(V_{2}-1)^{2}\\ +\pi_{31}\eta_{31}^{2}(\pi_{33}\eta_{33}^{2}+\pi_{22}\eta_{22}^{2}T_{2}^{2}V_{2 }^{2})\end{array}\Bigg{)}\end{array} \Bigg{)}}},\)\(f_{55}^{-1}=2m/t_{M}\) and 'T' indicate a transposition. As a result, the objective function can be determined by only \(f_{11}^{-1}\) because all terms in Equation (5) without \(f_{11}^{-1}\) are constant terms in this case. Nevertheless, the objective function is sufficiently complicated to obtain analytic optimal solutions. Therefore, a genetic algorithm (GA) was adopted to determine the practical PADT plans with two stress variables. Table 3 presents the practical PADT plans in various combined cases of the model parameters, where \(\pi_{33}=1-(\pi_{11}+\pi_{13}+\pi_{22}+\pi_{31})\). In Table 3, the parameters \(\delta_{2}\), \(\delta_{3}\), and \(\delta_{4}\) were chosen to examine the effect of parameters on the acceleration model. The parameters \(\delta_{2}\) and \(\delta_{3}\) indicate the severity of \(T\) (temperature) and \(V\) (voltage), respectively. In addition \(\delta_{4}\) represents the interaction effect of \(T\) and \(V\). That is, if \(\delta_{4}\) is positive or negative, there is an interaction between \(T\) and \(V\), whereas if \(\delta_{4}\) is zero, there is no interaction. In this study, the values of \(\delta_{4}\) are assumed to be \(-3\), \(0\), and \(3\). However, \(\delta_{1}\) was not considered in Table 3 since it is a constant term and does not affect the design of the practical PADT plans. In addition, some tendencies of the results of the practical PADT plan were derived from Table 3 as below. (1) As \(\delta_{2}\) increases, \(T_{2}\) increases and \(V_{2}\) decreases. (2) As \(\delta_{3}\) increases, \(T_{2}\) decreases and \(V_{2}\) increases. (3) As \(\delta_{2}\), \(\delta_{3}\), or \(\delta_{4}\) increases, \(\pi_{22}\) decreases. First, when the parameter \(\delta_{2}\) increases, \(T_{2}\) simultaneously increases with \(V_{2}\) decreasing. That is, if the mean degradation increment caused by temperature stress is greater than the mean degradation increment caused by voltage stress, the actual stress level \(T_{2}\) increases in the PADT plans. However, when \(T_{2}\) increases, \(V_{2}\) should be decreased by \(\delta_{4}\). Likewise, when \(\delta_{3}\) increases, \(V_{2}\) increases with \(T_{2}\) decreasing simultaneously. In addition, an increase in \(\delta_{2}\), \(\delta_{3}\), and/or \(\delta_{4}\) increases AF. When \(\delta_{2}\), \(\delta_{3}\), or \(\delta_{4}\) increases, more test units at \(\pi_{13}\), \(\pi_{31}\) and/or \(\pi_{33}\) are required corresponding to the parameters. As a result, \(\pi_{22}\) decreases. After the model parameters were determined, the practical PADT plan with three stress levels for two stress variables was designed in the second phase. Thus, the ratios and middle-stress levels (Table 3) used were: \(\pi_{11}=0.0082\), \(\pi_{13}=0.1590\), \(\pi_{22}=0.6032\), \(\pi_{31}=0.1923\), \(\pi_{33}=0.0373\), \(T_{2}=0.4737\) and \(V_{2}=0.7069\). According to the above results, four specimens were added to the design check (\(\pi_{11}\)). Additionally, 457, 557, and 92 specimens were allocated for the voltage limit test (\(\pi_{13}\)), the HTOL (\(\pi_{31}\)), and the HAST (\(\pi_{33}\)), respectively. A total of 1810 specimens were tested (\(\pi_{22}\)) under the middle-stress levels, 80.3 \({}^{\circ}\)C and 3.976 V, calculated based on Assumption 4. In the semiconductor area, qualification tests should be used to verify various item requirements. In particular, the HTOL and the HAST were included in the qualification tests, with 231 and 75 specimens, respectively, based on JEDEC standard JESD-47. Therefore, the specimens for the tests, as well as the reliability information, were more effectively applied during qualification.