Full Stress Tensor Measurement by Photoelasticity in Silicon Markus Stoehr 1 , Gerald Gerlach 2 , Thomas Härtling 2,3 , Stephan Schoenfelder 1 1 University of Applied Sciences Leipzig, Faculty of Engineering Sciences 2 Technische Universität Dresden, Institut für Festkörperelektronik 3 Fraunhofer Institute for Ceramic Technologies and Systems IKTS markus.stoehr@htwk-leipzig.de Summary: Photoelasticity offers a promising measurement tool for the in-line measurement of semiconductor ma- terials such as silicon. Photoelasticity is a contactless, optical and full-field measurement method based on stress-induced birefringence. However, it is in principle only capable of measuring stress differences and therefore not able to determine the quantitative stress state inside a material. In this work a method is presented to separate this stress difference using modified equations of the mechanical equilibrium in conjunction with the Finite-Difference-Method. This method takes into account the anisotropic photo- elastic law for mono-crystalline silicon and enables a full separation of the mechanical stresses with a single measurement. Keywords: photoelasticity, stress measurement method, full-field, contactless, semiconductors Introduction Photoelasticity is regarded a promising meas- urement principle to characterize stress fields in- side thin substrates. It is based on stress-in- duced birefringence, which causes a material to split an incoming light wave into two light waves each with different coefficients of refraction. In general, only the stress difference ሺ ߪ ଵଵ െ ߪ ଶଶ ሻ and the shear stresses ߪ ଵଶ can directly be meas- ured with this. However, a full stress tensor infor- mation gives the often more desirable full image of the stress state inside a material or compo- nent. In this work a method is developed to separate the full stress tensor information from a single photoelastic measurement. Since silicon shows a mechanically and photo-elastically anisotropic behavior, first the photo-elastical law for mono- crystalline silicon is developed. A general ap- proach and the explicit stress-optical relationship for several crystalline orientations are derived. With those, the stress differences and the shear stresses can be measured. Second, a method is developed to separate the stress difference into the single stress components by solving modi- fied equations of the mechanical equilibrium. These 2 nd -order partial differential equations are approximated by the Finite-Difference-Method. Applying FDM to the differential equations and using known initial values, the stress difference ሺ ߪ ଵଵ െ ߪ ଶଶ ሻ can further be separated into the sin- gle stress components ߪ ଵଵ and ߪ ଶଶ . This method is demonstrated for the full stress tensor determination of a silicon wafer under di- ametrical load and for the measurement of resid- ual stresses in a multi-crystalline silicon slab. Methods Photoelasticity can be described using a phe- nomenological approach suggested by Pockels [1, 2]. This model states that mechanical stresses lead to a change of the impermeability tensor ܤ ௜௝ which describes the optical properties. The change of impermeability is proportional to the stress tensor ߪ ௜௝ by the stress-optical tensor ߨ ௜௝௞௟ and adds to the unstressed impermeability ܤ ௜௝ ௢ : ܤ ௜௝ ൌ ܤ ௜௝ ௢ ൅ π ୧୨୩୪ ߪ ௞௟ (1) The change of impermeability depends also on the crystalline orientation and the load direction. Therefore, the stress-optical tensor has to be transformed accordingly. In this work, a general stress-optical relation and explicit relations for a (100), (110) and (111) crystalline orientation are developed. The neces- sary stress-optical material parameters ሺ ߨ ଵଵ െ ߨ ଵଶ ሻ and ߨ ସସ are determined using a (100) silicon wafer under diametrical load inside a grey-field polariscope. With this, the stress difference ሺ ߪ ଵଵ െ ߪ ଶଶ ሻ and the shear stress ߪ ଵଶ can be measured. To further separate this difference into its single stress SMSI 2021 Conference – Sensor and Measurement Science International 75 DOI 10.5162/SMSI2021/A6.1