edfas.org 31 ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 26 NO. 3 probability operator. The evaluation of the integral in equation 3 is not easy, because it represents a very small quantity and all the necessary information for the joint density function is not available. For these reasons, the first and the second order reliability meth- ods FORM/SORM have been developed.[10-15] For a given failure scenario, the reliability index β is evaluated by solving a constrained optimization problem (Fig. 6). The calculation of the reliability index is given by the following form: (Eq 4) In the FORM approximation, the failure probability is simply evaluated by: Pf ≈ Ф(-β) (Eq 5) where Ф is the standard Gaussian cumulated function. For practical engineering, equation 5 gives sufficiently accurate estimation of the failure probability. The design point P and the reliability index calculated with two methods FORM and SORM are presented in Table 4. It is clear that Young modulus (EX_AU) and thermal loading (Tf) affect the reliability and can lead to wire bond failure. The results obtained with the two methods are almost identical; the value of reliability index is small, which is equivalent to the probability of failure of 3.50%. Fig. 5 Shear and normal plastic strain distribution in the bond pad. Table 3 Statistical description of random variables Description Variables Mean Standard deviation Distribution Young’s modulus of wire bond, GPa EX_AU 90 6.35 Normal Young’s modulus of substrate, GPa EX_SUBS 300 20 Normal Thermal loading, °C Tf 500 20 Log-Normal Thickness of wire, μm Y_WIRE 25 0.65 Normal Thickness of pad, μm Y_PAD 25 0.65 Normal Thickness of substrate, μm Y_SUBS 625 16.5 Normal Table 4 Results of the reliability analysis of the assembly subjected to 25 to 500°C Description Variables Mean Design point (FORM) Design point (SORM) Young’s modulus of wire bond, GPa EX_AU 90 99.382 99.132 Thermal loading, °C Tf 500 506.86 505.91 Thickness of wire, μm Y_WIRE 25 24.831 24.831 Thickness of pad, μm Y_PAD 25 24.970 24.954 Reliability index Beta 1.804 1.848 Failure probability, % 3.559 3.225 Reliability level, % 96.44 96.77
RkJQdWJsaXNoZXIy MTYyMzk3NQ==