Modelling of silicon behaviour for wafer fracture assessment during rapid thermal annealing

Resumen

In the semiconductor manufacturing industry wafer handling introduces cracks at the wafer edge. During subsequent heat treatments these can propagate and cause wafer breakage during manufacture. The goal of this work is to predict wafer breakage during a given Rapid Thermal Annealing (RTA) from previous defects. For this purpose a methodology based on Finite Elements has been established to study the evolution of the stress fields during RTA around controlled cracks generated through indentation. Besides, in order to introduce temperature dependant silicon mechanical behaviour in the FEM model developed, Dislocation Dynamics (DD) simulations have been run. Aiming at simulating silicon behaviour, an open DD code, microMegas, has been modified by introducing different mobility laws for screw and mixed dislocations and by eliminating the line tension effect. Simulations have been run varying the temperature, the imposed strain rate and the initial dislocation density. The results obtained are in complete agreement with experimental measurements. Furthermore, they have served to check that the classical Alexander-Haasen model used to describe the dislocation multiplication in covalent crystals like silicon does not capture well dislocation multiplication, even though it gives reasonable results for the corresponding stress-strain curves. This has been the reason for proposing a new constitutive model for silicon. Controlled damage has been introduced through indentation at room temperature, both using Berkovich and Vickers tips. The stress fields around indentations have been calculated using FEM and a good correlation has been obtained between simulations and Raman measurements, even using isotropic plasticity to describe silicon behaviour. Furthermore, crack initiation and growth have been modelled through the usage of cohesive elements. For this task submodelling technique has been employed and the cohesive law has been calibrated by comparing simulations with experimentally measured crack lengths. The typical functional shape of the cohesive law has been changed in order to avoid convergence issues. Besides, it has been concluded that the only parameter with physical significance is the fracture energy. Finally a methodology based on FEM 'submodelling' technique has been developed for wafer fracture assessment during RTA processes. The J integral has been chosen as failure criteria and is computed from the results of a thermal simulation around an initial defect. It must be remarked that in order to define silicon's mechanical behaviour at different temperatures, the results obtained in DD simulations have been introduced in all the set of models. The procedure proposed allows for the determination of the critical value of J when data of a critical experiment is modelled. Even more, it also predicts failure under given conditions comparing the J value during the simulation with the critical J value, J_C. The critica J integral has been determined for two RTA processes, a 1073 K spike and a 1273 K plateau. According to experiments, the critical crack length for wafer failure was 2 and 8 mm respectively. The J_C values obtained are in accordance with experimental values found in bibliography. Furthermore, when the evolution of J integral is observed, it can be seen that maximum J value is achieved just after the plateau or after the spike, when the cooling has just begun in both types of treatments. This is also remarkable since it is in complete agreement with the experimental observations.