Parameter Identification in Stochastic Differential Equations to Model the Degradation of Metal Films

In automotive industry, reliability analysis of smart power semiconductor devices is mandatory for safety reasons. To reliably forecast the lifetime of devices, a mathematical description of the material degradation due to applied stress is required. In this work, the degradation process of a metal film of interest under thermo-mechanical stress is studied. Therefore, a bi-layer test structure (metal film on substrate) is used. Degradation refers to the formation, growth and coalescence of voids, initializing cracks that propagate through the metal film. Since the film's elastic-plastic deformation is triggered by random diffusional activities with drift, stochastic differential equations (SDEs) are used to model the evolution of the microstructural heterogeneity in the metallization.

The choice of an efficient numerical method to estimate the unknown parameters of the inverse problem is complex due to ill-posedness and rare measurements. To address these challenges, an approach by Dunker and Hohage of 2014 is followed, where parameter identification in SDE models is performed on basis of the Kolmogorov forward (Fokker-Planck) equation [1]. The mentioned approach has to be adapted to our application. Firstly, a stochastically consistent cost functional is defined, taking into account the different type of measurements and secondly, instead of estimating a function, a finite set of model parameters has to be recovered. For efficient computational optimization an adjoint approach is used to calculate the gradient of the cost functional. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)