Describing isotropic and anisotropic out-of-plane deformations in thin cubic materials by use of Zernike polynomials

Isotropic and anisotropic out-of-plane deformations induced by thin-film residual stress on thin cubic materials are studied. By transforming the compliance tensor, an analytical expression can be derived for the biaxial stiffness modulus for all directions in any given cubic crystal plane. A modified Stoneyʹs equation, including both isotropic and anisotropic terms, can be formulated to predict the anisotropic out-of-plane deformation. The isotropic and anisotropic deformations are then described using the Zernike polynomials U21 and U22, respectively. Experimental results from (100) and (110) silicon wafers confirm the model by quantitatively comparing the changes in Z21 and Z22 coefficients due to thin-film stress.