February_EDFA_Digital

edfas.org ELECTRONIC DEV ICE FA I LURE ANALYSIS | VOLUME 24 NO . 1 8 This hints at the suggestion that the crack probability is not normal, but Weibull-distributed. Assuming that, the force-dependent crack probability of multilayer stacks follows a two-parameteric Weibull distribution function P f of the form: with the Weibull modulus (or slope) m and the character- istic contact force F char . It can be seen from the curve of the distribution func- tion, that the maximum frequency of cracks occurs at a contact force of approximately 345 mN. By definition, this corresponds to the characteristic contact force F char with 63% crack probability and confirms the theory that critical forces causing cracks in brittle materials follow a Weibull distribution. Figure 10 shows the cumulative probability of first oxide cracks in Weibull scale as a function of the contact force. From this plot theWeibull modulus m is graphically derived from the linear regression line with a value of 18.3 and the characteristic contact force F char is 347 mN. According to statistical analysis, the distribution of the data points follows aWeibull distribution at a significance level of 0.05. At contact forces below 320 mN, however, the filtered data points are no closer to the linear regression line (green line) and are below the confidence band (red lines). The two-parametric Weibull distribution function does not accurately describe the crack probability here. The next section gives a hypothesis to improve the crack probability model. Fig. 9 Histogramof first oxide cracks for 100 contact cycles on test structureW11 using indenter FP10 as function of contact force, bin size 5 mN. Fig. 10 Cumulative probability of first oxide cracks for 100 contact cycles on test structure W11 using inden- ter FP10 in Weibull scale as function of contact force (linear regression line: green, 95% confidence band: red). HIGH-STATISTIC CRACK PROBABILITY ASSESSMENT FOR DIFFERENT LAYER STACKS To further prove the crack probability assessment for a higher sample size using the AE test method and to verify the two-parametricWeibull distributionmodel, two further AE indenter measurements were performed on test structure W06 and W08 using the indenter FP05 (tip diameter: 5 µm). For each test structure, 1000 successive contact cycles were performed at contact stress levels, which were high enough to reach a crack probability of approximately 100%. Following the experiments, a statisti- cal data analysiswas carriedout. The result of the cumula- tive crack probability in Weibull scale is shown in Fig. 11. Based on the results of the crack probabilitymeasure- ment on test structure W08, which has a 400 nm Cu layer below the upper SiO x layer, the characteristic contact force F char is 22.2 mN higher compared to test structure W06 (700 nm Cu layer). This result agrees well with the findings inUnterreitmeier, [1] stating that the characteristic contact force increaseswithdecreasingCu layer thickness. The Weibull modulus m for test structure W08 is smaller compared to structure W06, meaning that the scatter of the recorded AE crack signals for test structure W08 is lower compared to structure W06. Obviously a thicker Cu layer below the oxide layer is negatively affecting both the robustness and the reliability of the layer stack . Based on statistical calculations, the measurement points of both test structures follow a two-parametric Weibull distribution at a significance level of 0.05. For test