ADVANCED STATISTICAL ANALYSIS FOR TAIL-END PROBABILITY PREDICTION AND PERFORMANCE RESPONSE CALCULATION OF SEMICONDUCTOR PACKAGING PRODUCTS WITH A LARGE NUMBER OF INPUT VARIABLES

Abstract

Stochastic reliability modeling capabilities are developed and implemented for semiconductor packaging problems with a very large number of input variables (> 10 input variables). The capabilities are aimed at three critical areas in the semiconductor packaging product development: (1) prediction of tail-end probability (i.e., assembly yield loss) by advanced uncertainty propagation (UP) analyses, (2) determination of the statistical distributions of unknown design and/or manufacturing parameters by advanced statistical model calibrations, and (3) determination of the performance response of high-dimensional problems by developing an advanced metamodeling scheme.

In the first part, a comprehensive stochastic model is proposed and implemented to predict package-on-package (PoP) stacking yield loss based on non-contact open. The model takes into account all pad locations at the stacking interface while considering the statistical variations of warpages as well as solder ball and joint heights. The goal is achieved by employing (1) advanced approximate integration-based approach, called eigenvector dimension reduction (EDR) method, for the UP analysis; (2) the stress-strength interference (SSI), and (3) the union of events. The proposed approach is capable of handling the number of input variables much larger than that has been conceived as the practical limit of the UP analysis. The model can be used effectively to control the input uncertainties, and thus to achieve a yield goal for a given set of PoP designs.

In the second part, the unknown statistical distributions of two effective elastic properties of Sn-3.0Ag-0.5Cu solder joint of leadless chip resistors (LCRs), induced by an assembly condition, are determined by the advanced statistical model calibration. The UP analysis also utilizes the EDR method, which allows to take into account the statistical variations of six additional known input variables, including die thickness, solder joint height, termination length, and thickness and elastic moduli of a printed circuit board. The cyclic bending test results of LCR assemblies are used in conjunction with the maximum likelihood metric to obtain the statistical distributions of the effective properties. The cycles-to-failure distribution of the identical LCR assemblies subjected to a different loading level is predicted accurately by the calibrated model, which corroborates the validity of the proposed approach.

In the third part, an advanced metamodeling scheme, called partitioned bivariate Cut-high dimensional model representation (PB Cut-HDMR), is developed to consider the statistical correlation among input variables and to further reduce the computational burden encountered for high-dimensional problems without compromising accuracy. The statistical correlation is handled by eigen-decomposition of a covariance matrix. The latter is achieved by the HDMR-factorial design (HDMR-FD) hybrid method. The validity of the proposed scheme is verified by comparing the performance of the proposed scheme with the full bivariate Cut-HDMR. The proposed scheme is implemented successfully to construct an accurate metamodel for a problem with 12 input variables among which 2 pairs are correlated.